**ALGEBRAIC NUMBER THEORY**

J.S. MILNE

Abstract. These are the notes for a course taught at the University of Michigan
in F92 as Math 676. They are available atwww.math.lsa.umich.edu/*∼*jmilne/.

Please send comments and corrections to me at jmilne@umich.edu.

v2.01 (August 14, 1996.) First version on the web.

v2.10 (August 31, 1998.) Fixed many minor errors; added exercises and index.

Contents

**Introduction**. . . .1
The ring of integers 1; Factorization 2; Units 4; Applications 5; A brief
history of numbers 6; References. 7.

**1. Preliminaries from Commutative Algebra**. . . 10
Basic deﬁnitions 10; Noetherian rings 10; Local rings 12; Rings of fractions
12; The Chinese remainder theorem 14; Review of tensor products 15;

Extension of scalars 17; Tensor products of algebras 17; Tensor products of ﬁelds 17.

**2. Rings of Integers**. . . 19
Symmetric polynomials 19; Integral elements 20; Review of bases of *A*-
modules 25; Review of norms and traces 25; Review of bilinear forms 26;

Discriminants 26; Rings of integers are ﬁnitely generated 28; Finding the ring of integers 30; Algorithms for ﬁnding the ring of integers 33.

**3. Dedekind Domains; Factorization**. . . 37
Discrete valuation rings 37; Dedekind domains 38; Unique factorization
39; The ideal class group 43; Discrete valuations 46; Integral closures of
Dedekind domains 47; Modules over Dedekind domains (sketch). 48; Fac-
torization in extensions 49; The primes that ramify 50; Finding factoriza-
tions 53; Examples of factorizations 54; Eisenstein extensions 56.

**4. The Finiteness of the Class Number**. . . 58
Norms of ideals 58; Statement of the main theorem and its consequences
59; Lattices 62; Some calculus 67; Finiteness of the class number 69; Binary
quadratic forms 71;

**5. The Unit Theorem**. . . 7 3
Statement of the theorem 73; Proof that *U**K* is ﬁnitely generated 74; Com-
putation of the rank 75; *S*-units 77; Finding fundamental units in real

c1996, 1998, J.S. Milne. You may make one copy of these notes for your own personal use.

i

0 J.S. MILNE

quadratic ﬁelds 77; Units in cubic ﬁelds with negative discriminant 78;

Finding *µ*(*K*) 80; Finding a system of fundamental units 80; Regulators
80;

**6. Cyclotomic Extensions; Fermat’s Last Theorem.**. . . .82
The basic results 82; Class numbers of cyclotomic ﬁelds 87; Units in cyclo-
tomic ﬁelds 87; Fermat’s last theorem 88;

**7. Valuations; Local Fields**. . . 91
Valuations 91; Nonarchimedean valuations 91; Equivalent valuations 93;

Properties of discrete valuations 95; Complete list of valuations for Q 95;

The primes of a number ﬁeld 97; Notations 97; Completions 98; Com-
pletions in the nonarchimedean case 99; Newton’s lemma 102; Extensions
of nonarchimedean valuations 105; Newton’s polygon 107; Locally compact
ﬁelds 108; Unramiﬁed extensions of a local ﬁeld 109; Totally ramiﬁed exten-
sions of*K* 111; Ramiﬁcation groups 112; Krasner’s lemma and applications
113; A Brief Introduction to PARI 115.

**8. Global Fields**. . . 116
Extending valuations 116; The product formula 118; Decomposition groups
119; The Frobenius element 121; Examples 122; Application: the quadratic
reciprocity law 123; Computing Galois groups (the hard way) 123; Comput-
ing Galois groups (the easy way) 124; Cubic polynomials 126; Chebotarev
density theorem 126; Applications of the Chebotarev density theorem 128;

Topics not covered 130; More algorithms 130; The Hasse principle for qua- dratic forms 130; Algebraic function ﬁelds 130.

**Exercises**. . . .132.

It is standard to use Gothic (fraktur) letters for ideals:

a b c m n p q A B C M N P Q
*a b c m n p q A B* *C* *M* *N* *P* *Q*
I use the following notations:

*X* *≈Y* *X* and *Y* are isomorphic;

*X* *∼*=*Y* *X* and *Y* are canonically isomorphic
or there is a given or unique isomorphism;

*X* =^{df} *Y* *X* is deﬁned to be *Y*, or equals*Y* by deﬁnition;

*X* *⊂Y* *X* is a subset of *Y* (not necessarily proper).

Introduction 1

Introduction

An *algebraic number ﬁeld* is a ﬁnite extension of Q; an *algebraic number* is an
element of an algebraic number ﬁeld. Algebraic number theory studies the arithmetic
of algebraic number ﬁelds — the ring of integers in the number ﬁeld, the ideals in
the ring of integers, the units, the extent to which the ring of integers fails to be
have unique factorization, and so on. One important tool for this is “localization”, in
which we complete the number ﬁeld relative to a metric attached to a prime ideal of
the number ﬁeld. The completed ﬁeld is called a*local ﬁeld* — its arithmetic is much
simpler than that of the number ﬁeld, and sometimes we can answer questions by
ﬁrst solving them locally, that is, in the local ﬁelds.

An*abelian extension* of a ﬁeld is a Galois extension of the ﬁeld with abelian Galois
group. Global class ﬁeld theory classiﬁes the abelian extensions of a number ﬁeld *K*
in terms of the arithmetic of*K*; local class ﬁeld theory does the same for local ﬁelds.

This course is concerned with algebraic number theory. Its sequel is on class ﬁeld theory (see my notes CFT).

I now give a quick sketch of what the course will cover. The*fundamental theorem of*
*arithmetic* says that integers can be uniquely factored into products of prime powers:

an *m* = 0 in Zcan be written in the form,

*m* =*up*^{r}_{1}^{1}*· · ·p*^{r}_{n}^{n}*,* *u*=*±*1*,* *p*_{i} prime number,*r*_{i} *>*0*,*
and this factorization is essentially unique.

Consider more generally an integral domain *A*. An element *a* *∈A* is said to be a
*unit* if it has an inverse in*A*; I write*A*^{×} for the multiplicative group of units in *A*.

An element*p*of *A* is said to*prime* if it is neither zero nor a unit*,* and if
*p|ab⇒p|a* or *p|b.*

If *A* is a principal ideal domain, then every nonzero nonunit element*a* of *A* can be
written in the form,

*a*=*p*^{r}_{1}^{1}*· · ·p*^{r}_{n}^{n}*,* *p*_{i} prime element, *r*_{i} *>*0*,*

and the factorization is unique up to order and replacing each *p*_{i} with an associate,
i.e., with its product with a unit.

Our ﬁrst task will be to discover to what extent unique factorization holds, or fails
to hold, in number ﬁelds. Three problems present themselves. First, factorization in
a ﬁeld only makes sense with respect to a subring, and so we must deﬁne the “ring
of integers” *O**K* in our number ﬁeld*K*. Secondly, since unique factorization will in
general fail, we shall need to ﬁnd a way of measuring by how much it fails. Finally,
since factorization is only considered up to units, in order to fully understand the
arithmetic of*K*, we need to understand the structure of the group of units*U*_{K} in*O**K*.
Resolving these three problems will occupy the ﬁrst ﬁve sections of the course.

*The ring of integers.* Let *K* be an algebraic number ﬁeld. Because *K* is of ﬁnite
degree overQ, every element*α* of *K* is a root of a monic polynomial

*f*(*X*) =*X*^{n}+*a*_{1}*X*^{n}^{−}^{1} +*· · ·*+*a*_{0}*,* *a*_{i} *∈*Q*.*

2 Introduction

If *α* is a root of a monic polynomial with *integer* coeﬃcients*,* then *α* is called an
*algebraic integer* of*K*. We shall see that the algebraic integers form a subring *O**K* of
*K*.

The criterion as stated is diﬃcult to apply. We shall see that to prove that *α* is
an algebraic integer, it suﬃces to check that its minimum polynomial (relative toQ)
has integer coeﬃcients.

Consider for example the ﬁeld *K* = Q[

*d*], where *d* is a square-free integer. The
minimum polynomial of *α* =*a*+*b√*

*d*,*b* = 0, *a, b∈*Q, is
(*X−*(*a*+*b√*

*d*))(*X−*(*a−b√*

*d*)) =*X*^{2}*−*2*aX* + (*a*^{2}*−b*^{2}*d*)*.*

Thus *α* is an algebraic integer if and only if

2*a∈*Z*,* *a*^{2}*−b*^{2}*d∈*Z*.*
From this it follows easily that

*O**K* =Z[*√*

*d*] =*{m*+*n√*

*d|m, n∈*Z*}* if *d≡*2*,*3 mod 4*,*
and

*O**K* =*{m*+*n*1 +*√*
*d*

2 *|m, n∈*Z*}* if *d≡*1 mod 4*,*
i.e., *O**K* is the set of sums *m*^{} +*n*^{}*√*

*d* with *m*^{} and *n*^{} either both integers or both
half-integers.

Let*ζ*_{d}be a primitive*d*^{th} root of 1, for example,*ζ*_{d} = exp(2*πi/d*), and let*K* =Q[*ζ*_{d}].

Then we shall see that

*O**K* =Z[*ζ**d*] = *{*

*m**i**ζ*_{d}^{i} *|m**i* *∈*Z*}.*
as one would hope.

*Factorization.* An element *p* of an integral domain *A* is said to be *irreducible* if it
is neither zero nor a unit, and can’t be written as a product of two nonunits. For
example, a prime element is (obviously) irreducible. A ring*A*is a*unique factorization*
*domain* if every nonzero nonunit element of *A* can be expressed as a product of
irreducible elements in essentially one way. Is *O**K* a unique factorization domain?

No, not in general!

In fact, we shall see that each element of *O**K* can be written as a product of
irreducible elements (this is true for all Noetherian rings) — it is the uniqueness that
fails.

For example, in Z[*√*

*−*5] we have

6 = 2*·*3 = (1 +*√*

*−*5)(1*−√*

*−*5)*.*

To see that 2, 3, 1 +*√*

*−*5, 1*−√*

*−*5 are irreducible, and no two are associates, we use
the norm map

Nm :Q[*√*

*−*5]*→*Q*,* *a*+*b√*

*−*5*→a*^{2}+ 5*b*^{2}*.*
For*α* *∈ O**K*, we have

Nm(*α*) = 1 *⇐⇒* *αα*¯ = 1 *⇐⇒* *α* is a unit. (*)

Introduction 3

If 1 +*√*

*−*5 =*αβ*, then Nm(*αβ*) = Nm(1 +*√*

*−*5) = 6. Thus Nm(*α*) = 1*,*2*,*3, or 6. In
the ﬁrst case, *α* is a unit, the second and third cases don’t occur, and in the fourth
case *β* is a unit. A similar argument shows that 2*,*3, and 1*−√*

*−*5 are irreducible.

Next note that (*) implies that associates have the same norm, and so it remains to
show that 1 +*√*

*−*5 and 1*−√*

*−*5 are not associates, but
1 +*√*

*−*5 = (*a*+*b√*

*−*5)(1*−√*

*−*5)
has no solution with *a, b∈*Z.

Why does unique factorization fail in*O**K*? The problem is that irreducible elements
in*O**K* need not be prime. In the above example, 1 +*√*

*−*5 divides 2*·*3 but it divides
neither 2 nor 3. In fact, in an integral domain in which factorizations exist (e.g. a
Noetherian ring), factorization is unique if all irreducible elements are prime.

What can we recover? Consider

210 = 6*·*35 = 10*·*21*.*

If we were naive, we might say this shows factorization is not unique inZ; instead, we recognize that there is a unique factorization underlying these two decompositions, namely,

210 = (2*·*3)(5*·*7 ) = (2*·*5)(3*·*7)*.*

The idea of Kummer and Dedekind was to enlarge the set of “prime numbers” so
that, for example, in Z[*√*

*−*5] there is a unique factorization,
6 = (p1*·*p2)(p3*·*p4) = (p1*·*p3)(p2*·*p4)*,*

underlying the above factorization; here thep*i* are “ideal prime factors”.

How do we deﬁne “ideal factors”? Clearly, an ideal factor should be character- ized by the algebraic integers it divides. Moreover divisibility by a should have the following properties:

a*|*0; a*|a,*a*|b⇒*a*|a±b*; a*|a⇒*a*|ab* for all *b* *∈ O**K**.*
If in addition division by ahas the property that

a*|ab⇒*a*|a* or a*|b,*

then we call aa “prime ideal factor”. Since all we know about an ideal factor is the
set of elements it divides, we may as well identify it with this set. Thus an ideal
factor is a set of elementsa*⊂ O**K* such that

0*∈*a; *a, b∈*a*⇒a±b* *∈*a; *a∈*a*⇒ab∈*a for all *b∈ O**K*;
it is prime if an addition,

*ab∈*a*⇒a* *∈*aor *b∈*a*.*

Many of you will recognize that an ideal factor is what we now call an *ideal*, and a
prime ideal factor is a *prime ideal*.

There is an obvious notion of the product of two ideals:

ab*|c* *⇐⇒* *c*=

*a**i**b**i**,* a*|a**i**,* b*|b**i**.*
In other words,

ab=*{*

*a*_{i}*b*_{i} *|a*_{i} *∈*a*,* *b*_{i} *∈*b*}.*

4 Introduction

One see easily that this is again an ideal, and that if

a= (*a*_{1}*, ..., a*_{m}) andb= (*b*_{1}*, ..., b*_{n})
then

a*·*b = (*a*_{1}*b*_{1}*, a*_{1}*b*_{2}*, ..., a*_{i}*b*_{j}*, ..., a*_{m}*b*_{n})*.*

With these deﬁnitions, one recovers unique factorization: if*a*= 0, then there is an
essentially unique factorization:

(*a*) =p^{r}1^{1}*· · ·*p^{r}*n*^{n} with eachp*i* a prime ideal.

In the above example,
(6) = (2*,*1 +*√*

*−*5)(2*,*1*−√*

*−*5)(3*,*1 +*√*

*−*5)(3*,*1*−√*

*−*5)*.*

In fact, I claim

(2*,*1 +*√*

*−*5)(2*,*1*−√*

*−*5) = (2)
(3*,*1 +*√*

*−*5)(3*,*1*−√*

*−*5) = (3)
(2*,*1 +*√*

*−*5)(3*,*1 +*√*

*−*5) = (1 +*√*

*−*5)
(2*,*1*−√*

*−*5)(3*,*1*−√*

*−*5) = (1*−√*

*−*5)*.*

For example, (2*,*1 +*√*

*−*5)(2*,*1*−√*

*−*5) = (4*,*2 + 2*√*

*−*5*,*2*−*2*√*

*−*5*,*6). Since every
generator is divisible by 2, (2*,*1 +*√*

*−*5)(2*,*1*−√*

*−*5)*⊂*(2). Conversely,
2 = 6*−*4*∈*(4*,*2 + 2*√*

*−*5*,*2*−*2*√*

*−*5*,*6)
and so (2*,*1 +*√*

*−*5)(2*,*1*−√*

*−*5) = (2). Moreover, the four ideals (2*,*1 + *√*

*−*5),
(2*,*1*−√*

*−*5), (3*,*1 +*√*

*−*5), and (3*,*1*−√*

*−*5) are all prime. For example
Z[*√*

*−*5]*/*(3*,*1*−√*

*−*5) =Z*/*(3)*,*
which is an integral domain.

How far is this from what we want, namely, unique factorization of elements? In
other words, how many “ideal” elements have we had to add to our “real” elements
to get unique factorization. In a certain sense, only a ﬁnite number: we shall see
that there is a ﬁnite set of idealsa1*, ...,*a*h* such that every ideal is of the form a*i**·*(*a*)
for some *i* and some *a* *∈ O**K*. Better, we shall construct a group *I* of “fractional”

ideals in which the principal fractional ideals (*a*),*a∈K*^{×}, form a subgroup*P* of ﬁnite
index. The index is called the *class number* *h*_{K} of *K*. We shall see that

*h*_{K} = 1 *⇐⇒ O**K* is a principal ideal domain *⇐⇒ O**K* is a unique factorization domain.

*Units.* Unlike Z, *O**K* can have an inﬁnite number of units. For example, (1 +*√*
2) is
a unit of inﬁnite order in Z[*√*

2] :
(1 +*√*

2)(*−*1 +*√*

2) = 1; (1 +*√*

2)^{m} = 1 for *m≥*1*.*

In fact Z[*√*

2]^{×} =*{±*(1 +*√*

2)^{m} *|m∈*Z*}*, and so
Z[*√*

2]^{×}*≈ {±*1*} × {*free abelian group of rank 1*}.*

Introduction 5

In general, we shall show (unit theorem) that the roots of 1 in *K* form a ﬁnite group
*µ*(*K*), and that

*O**K*^{×} *≈µ*(*K*)*×*Z^{r} (as an abelian group);

moreover, we shall ﬁnd *r.*

*Applications.* I hope to give some applications. One motivation for the development
of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e.,
that there are no integer solutions to the equation

*X*^{m}+*Y*^{m} =*Z*^{m}
when *m≥*3, except for the obvious solutions.

When *m* = 3, this can proved by the method of “inﬁnite descent”, i.e., from
one solution, you show that you can construct a smaller solution, which leads to a
contradiction^{1}. The proof makes use of the factorization

*Y*^{3} =*Z*^{3}*−X*^{3} = (*Z−X*)(*Z*^{2}+*XZ*+*X*^{2})*,*

and it was recognized that a stumbling block to proving the theorem for larger *m* is
that no such factorization exists into polynomials with integer coeﬃcients. This led
people to look at more general factorizations.

In a very famous incident, the French mathematician Lam´e gave a talk at the
Paris Academy in 1847in which he claimed to prove Fermat’s last theorem using the
following ideas. Let *p >*2 be a prime, and suppose *x*,*y*, *z* are nonzero integers such
that

*x*^{p}+*y*^{p} =*z*^{p}*.*
Write

*x*^{p} =*z*^{p}*−y*^{p} =

(*z−ζ*^{i}*y*)*,* 0*≤i≤p−*1*,* *ζ* =*e*^{2πi/p}*.*

He then showed how to obtain a smaller solution to the equation, and hence a contra-
diction. Liouville immediately questioned a step in Lam´e’s proof in which he assumed
that, in order to show that each factor (*z* *−ζ*^{i}*y*) is a *p*^{th} power, it suﬃces to show
that the factors are relatively prime in pairs and their product is a *p*^{th} power. In
fact, Lam´e couldn’t justify his step (Z[*ζ*] is not always a principal ideal domain), and
Fermat’s last theorem remains unproven to the present day^{2}. However, shortly after
Lam´e’s embarrassing lecture, Kummer used his results on the arithmetic of the ﬁelds
Q[*ζ*] to prove Fermat’s last theorem for all “regular primes”.

Another application is to ﬁnding Galois groups. The splitting ﬁeld of a polynomial
*f*(*X*) *∈* Q[*X*] is a Galois extension of Q. In the basic graduate algebra course (see
FT), we learn how to compute the Galois group only when the degree is very small
(e.g., *≤* 3). By using algebraic number theory one can write down an algorithm to
do it for any degree.

1The simplest proof by inﬁnite descent is that showing that *√*

2 is irrational.

2Written in 1992.

6 Introduction

**A brief history of numbers.** Prehistory(??-1600). Basic arithmetic was devel-
oped in many parts of the world thousands of years ago. For example, 3,500 years
ago the Babylonians apparently knew how to construct the solutions to

*X*^{2}+*Y*^{2} =*Z*^{2}*.*
At least they knew that

(4961)^{2} + (6480)^{2} = (8161)^{2}

which could scarcely be found by trial and error. The Chinese remainder theorem was known in China, thousands of years ago. The Greeks knew the fundamental theorem of arithmetic, and, of course, Euclid’s algorithm.

Fermat (1601–1665). Apart from his famous last “theorem”, he invented the method of inﬁnite descent. He also posed the problem of ﬁnding integer solutions to the equation,

*X*^{2}*−AY*^{2} = 1*,* *A* *∈*Z*,* (*)
which is essentially the problem^{3} of ﬁnding the units in Z[*√*

*A*]. The English math-
ematicians found an algorithm for solving the problem, but neglected to show that
the algorithm always works.

Euler(1707–1783). Among many other works, he discovered the quadratic reci- procity law.

Lagrange(1736–1813). He proved that the algorithm for solving (*) always leads to a solution.

Legendre (1752–1833). He proved the “Hasse principle” for quadratic forms in
three variables over Q: the quadratic form *Q*(*X, Y, Z*) has a nontrivial zero in Q if
and only if it has one inRand the congruence*Q≡*0 mod*p*^{n} has a nontrivial solution
for all*p* and *n*.

Gauss(1777–1855). He found many proofs of the quadratic reciprocity law:

*p*
*q*

*q*
*p*

= (*−*1)^{(p}^{−}^{1)(q}^{−}^{1)/4}*,* *p, q* odd primes.

He studied the Gaussian integers Z[*i*] in order to ﬁnd a quartic reciprocity law. He
studied the classiﬁcation of binary quadratic forms over Z which, as we shall see, is
closely related to the problem of ﬁnding the class numbers of quadratic ﬁelds.

Dirichlet (1805–1859). He proved the following “unit theorem”: let*α* be a root
of a monic irreducible polynomial *f*(*X*) with integer coeﬃcients; suppose that *f*(*X*)
has *r* real roots and 2*s* complex roots; then Z[*α*]^{×} is a ﬁnitely generated group of
rank *r*+*s−*1. He proved a famous analytic formula for the class number.

Kummer(1810–1893). He made a deep study of the arithmetic of cyclotomic ﬁelds, motivated by a search for higher reciprocity laws. His general result on Fermat’s last theorem is the most important to date.

Hermite (1822–1901).

Eisenstein (1823–1852).

3The Indian mathematician Bhaskara (12th century) knew general rules for ﬁnding solutions to the equation.

Introduction 7

Kronecker (1823–1891). He developed an alternative to Dedekind’s ideals. He
also had one of the most beautiful ideas in mathematics, the *Kronecker liebster Ju-*
*gendtraum*, for generating abelian extensions of number ﬁelds.

Riemann (1826–1866). Made the Riemann hypothesis.

Dedekind (1831–1916). He was the ﬁrst mathematician to formally deﬁne ﬁelds

— many of the basic theorems on ﬁelds in basic graduate algebra courses were proved by him. He also found the correct general deﬁnition of the ring of integers in a number ﬁeld, and he proved that ideals factor uniquely into products of prime ideals.

Moreover, he improved the Dirichlet unit theorem.

Weber (1842–1913). Made important progress in class ﬁeld theory and the Kro- necker Jugendtraum.

Hensel(1861–1941). He introduced the notion of the*p*-adic completion of a ﬁeld.

Hilbert(1862–1943). He wrote a very inﬂuential book on algebraic number theory in 1897, which gave the ﬁrst systematic account of the theory. Some of his famous problems were on number theory, and have also been inﬂuential.

Takagi(1875–1960). He made very important advances in class ﬁeld theory.

Hecke(1887–1947). Introduced Hecke *L*-series.

Artin (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class ﬁeld theory.

Hasse(1898–1979). Proved the Hasse principle for all quadratic forms over number ﬁelds.

Weil (1906–1998). Deﬁned the Weil group, which enabled him to give a common
generalization of Artin *L*-series and Hecke*L*-series.

Chevalley (1909–??). The main statements of class ﬁeld theory are purely al- gebraic, but all the earlier proofs used analysis. Chevalley gave a purely algebraic proof.

Iwasawa(1917– ). He introduced an important new approach into the study of algebraic number theory which was suggested by the theory of curves over ﬁnite ﬁelds.

Tate (1925– ). With Artin, he gave a complete cohomological treatment of class ﬁeld theory. With Lubin he introduced a concrete way of generating abelian exten- sions of local ﬁelds.

Langlands(1936– ). “Langlands’s philosophy” is a vast series of conjectures that,
among other things, contains a *nonabelian* class ﬁeld theory.

**References.** *Books on algebraic number theory.*

Artin, E., *Theory of Algebraic Numbers*, G¨ottingen notes, 1959. Elegant; good exam-
ples; but he adopts a valuation approach rather than the ideal-theoretic approach we
use in this course.

Artin, E.,*Algebraic Numbers and Algebraic Functions*, Nelson, 1968. Covers both the
number ﬁeld and function ﬁeld case.

Borevich, Z. I., and Shafarevich, I. R., *Number Theory*, Academic Press, 1966.

In addition to basic algebraic number theory, it contains material on diophantine equations.

8 Introduction

Cassels, J.W.S., and Fr¨ohlich, A., Eds., *Algebraic Number Theory*, Academic Press,
1967. The proceedings of an instructional conference. Many of the articles are excel-
lent, for example, those of Serre and Tate on class ﬁeld theory.

Cassels, J.W.S.,*Local ﬁelds*, London Math. Soc., 1986. Concentrates on local ﬁelds,
but does also deal with number ﬁelds, and it gives some interesting applications.

Cohn, P.M., *Algebraic Numbers and Algebraic Functions*, Chapman and Hall, 1991.

The valuation approach.

Dedekind, R.,*Theory of Algebraic Integers*, Cambridge Univ. Press, 1996 (translation
of the 1877 French original). Develops the basic theory through the ﬁniteness of the
class number in a way that is surprising close to modern approach in, for example,
these notes.

Edwards, H., *Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number*
*Theory*, Springer, 1977. A history of algebraic number theory, concentrating on the
eﬀorts to prove Fermat’s last theorem. Edwards is one of the most reliable writers on
the history of number theory.

Fr¨ohlich, A., and Taylor, M.J., *Algebraic Number Theory*, Cambridge Univ. Press,
1991. Lots of good problems.

Goldstein, L.J., *Analytic Number Theory*, Prentice-Hall, 1971. Similar approach to
Lang 1970, but the writing is a bit careless. Sometimes includes more details than
Lang, and so it is probably easier to read.

Janusz, G. *Algebraic Number Fields*, Second Edn, Amer. Math. Soc., 1996. It covers
both algebraic number theory and class ﬁeld theory, which it treats from a lowbrow
analytic/algebraic approach. In the past, I sometimes used the ﬁrst edition as a text
for this course and its sequel.

Lang, S. *Algebraic Numbers Theory*, Addison-Wesley, 1970. Diﬃcult to read unless
you already know it, but it does contain an enormous amount of material. Covers alge-
braic number theory, and it does class ﬁeld theory from a highbrow analytic/algebraic
approach.

Marcus, D. *Number Fields*, Springer, 1977. This is a rather pleasant down-to-earth
introduction to algebraic number theory.

Narkiewicz, W. *Algebraic Numbers*, Springer, 1990. Encyclopedic coverage of alge-
braic number theory.

Samuel, P.,*Algebraic Theory of Numbers*, Houghton Miﬄin, 1970. A very easy treat-
ment, with lots of good examples, but doesn’t go very far.

Serre, J.-P.*Corps Locaux*, Hermann, 1962 (Translated as Local Fields). A classic. An
excellent account of local ﬁelds, cohomology of groups, and local class ﬁeld theory.

The local class ﬁeld theory is bit dated (Lubin-Tate groups weren’t known when the book was written) but this is the best book for the other two topics.

Weil, A.,*Basic Number Theory*, Springer, 1967. Very heavy going, but you will learn
a lot if you manage to read it (covers algebraic number theory and class ﬁeld theory).

Weiss, R.,*Algebraic Number Theory*, McGraw-Hill, 1963. Very detailed; in fact a bit
too fussy and pedantic.

9

Weyl, H.,*Algebraic Theory of Numbers*, Princeton Univ. Press, 1940. One of the ﬁrst
books in English; by one of the great mathematicians of the twentieth century. Id-
iosyncratic — Weyl prefers Kronecker to Dedekind, e.g., see the section “Our disbelief
in ideals”.

*Computational Number Theory.*

Cohen, H.,*A Course in Computational Number Theory*, Springer, 1993.

Lenstra, H., *Algorithms in Algebraic Number Theory*, Bull. Amer. Math. Soc., 26,
1992, 211–244.

Pohst and Zassenhaus, *Algorithmic Algebraic Number Theory*, Cambridge Univ.

Press, 1989.

The two books provide algorithms for most of the constructions we make in this course. The ﬁrst assumes the reader knows number theory, whereas the second de- velops the whole subject algorithmically. Cohen’s book is the more useful as a sup- plement to this course, but wasn’t available when these notes were ﬁrst written, and so the references are to Pohst and Zassenhaus. While the books are concerned with more-or-less practical algorithms for ﬁelds of small degree and small discriminant, Lenstra’s article concentrates on ﬁnding “good” general algorithms.

*Additional references*

Atiyah, M.F., and MacDonald, I.G.,*Introduction to Commutative Algebra*, Addison-
Wesley, 1969. I use this as a reference on commutative algebra.

Washington, L., *Introduction to Cyclotomic Fields*, 1982. This is the best book on
cyclotomic ﬁelds.

I will sometimes refer to my other course notes:

**GT:** Group Theory (594)

**FT:** Fields and Galois Theory (594)
**EC:** Elliptic Curves (679).

**CFT:** Class Field Theory (776).

10

1. Preliminaries from Commutative Algebra

Many results that were ﬁrst proved for rings of integers in number ﬁelds are true for more general commutative rings, and it is more natural to prove them in that context.

**Basic definitions.** All rings will be commutative, and have an identity element (i.e.,
an element 1 such that 1*a* =*a* for all *a∈A*), and a homomorphism of rings will map
the identity element to the identity element.

A ring *B* together with a homomorphism of rings *A→B* will be referred to as an
*A*-*algebra*. We use this terminology mainly when*A*is a subring of*B*. In this case, for
elements*β*_{1}*, ..., β*_{m}of *B*,*A*[*β*_{1}*, ..., β*_{m}] denotes the smallest subring of*B* containing*A*
and the *β*_{i}. It consists of all polynomials in the *β*_{i} with coeﬃcients in*A*, i.e., elements
of the form

*a**i*1*...i**m**β*_{1}^{i}^{1}*...β*_{m}^{i}^{m}*,* *a**i*1*...i**m* *∈A.*

We also refer to*A*[*β*_{1}*, ..., β*_{m}] as the*A*-subalgebra of*B* *generated* by the*β*_{i}, and when
*B* =*A*[*β*_{1}*, ..., β*_{m}] we say that the *β*_{i} *generate* *B* as an *A*-algebra.

For elements*a*_{1}*, a*_{2}*, . . .* of *A*, (*a*_{1}*, a*_{2}*, . . .*) denotes the smallest ideal containing the
*a**i*. It consists of ﬁnite sums

*c**i**a**i*, *c**i* *∈* *A*, and it is called the *ideal generated by*
*a*1*, a*2*, . . .*. When a and b are ideals in*A*, we deﬁne

a+b=*{a*+*b|a* *∈*a, *b* *∈*b*}.*

It is again an ideal in*A* — in fact, it is the smallest ideal containing both aand b. If
a= (*a*1*, ..., a**m*) and b= (*b*1*, ..., b**n*), thena+b= (*a*1*, ..., a**m**, b*1*, ..., b**n*)*.*

Given an ideal a in *A*, we can form the quotient ring *A/*a. Let *f*: *A* *→* *A/*a be
the homomorphism *a→a*+a; thenb *→f*^{−}^{1}(b) deﬁnes a one-to-one correspondence
between the ideals of *A/*a and the ideals of *A* containinga, and

*A/f*^{−1}(b)*→*^{≈} (*A/*a)*/*b*.*

A proper ideal a of *A* is *prime* if *ab* *∈*a *⇒a* or *b* *∈* a. An ideal a is prime if and
only if the quotient ring *A/*a is an integral domain. An element*p* of *A* is said to be
*prime* if (*p*) is a prime ideal; equivalently, if *p|ab⇒p|a* or *p|b*.

A proper ideala in*A* is *maximal* if there does not exist an ideal b,ab *A.*An
idealais maximal if and only if*A/*a is a ﬁeld. Every proper idealaof *A*is contained
in a maximal ideal — if *A* is Noetherian (see below) this is obvious; otherwise the
proof requires Zorn’s lemma. In particular, every nonunit in *A* is contained in a
maximal ideal.

There are the implications: *A* is a Euclidean domain *⇒* *A* is a principal ideal
domain*⇒A*is a unique factorization domain (see any good graduate algebra course).

**Noetherian rings.**

Lemma 1.1. *The following conditions on a ring* *A* *are equivalent:*

(a) *Every ideal in* *A* *is ﬁnitely generated.*

1. Preliminaries from Commutative Algebra 11

(b) *Every ascending chain of ideals*

a1 *⊂*a2 *⊂ · · · ⊂*a*n**⊂ · · ·*

*becomes stationary, i.e., after a certain point* a*n* =a*n*+1 =*· · ·.*

(c) *every nonempty set* *S* *of ideals in* *A* *has a maximal element* a*, i.e., there is an*
*ideal* a *in* *S* *that is not contained in any other ideal in* *S.*

Proof. (a)*⇒*(b): Let a= *∪*a*i*; it is an ideal, and hence is ﬁnitely generated, say
a= (*a*_{1}*, . . . , a*_{r}). For some *n*,a*n* will contain all the *a*_{i}, and soa*n* =a*n*+1 =*· · ·*=a.
(b)*⇒*(a): Consider an ideala. If a= (0), thenais generated by the empty set, which
is ﬁnite. Otherwise there is an element*a*_{1} *∈*a, *a*_{1} = 0. Ifa= (*a*_{1}), thenais certainly
ﬁnitely generated. If not, there is an element *a*2 *∈* a such that (*a*1) (*a*1*, a*2).

Continuing in this way, we obtain a chain of ideals
(*a*1)(*a*1*, a*2)*· · ·* *.*
This process must eventually stop with (*a*_{1}*, . . . , a*_{n}) =a.

(b)*⇒*(c): Let a1 *∈* *S*. If a1 is not a maximal element of *S*, then there is an a2 *∈* *S*
such that a1 a2. Ifa2 is not maximal, then there is an a3 etc.. From (b) we know
that this process will lead to a maximal element after only ﬁnitely many steps.

(c)*⇒*(b): Apply (c) to the set *S*=*{*a1*,*a2*, . . .}*.

A ring*A*satisfying the equivalent conditions of the lemma is said to be*Noetherian*^{4}
A famous theorem of Hilbert states that *k*[*X*_{1}*, ..., X*_{n}] is Noetherian. In practice,
almost all the rings that arise naturally in algebraic number theory or algebraic geom-
etry are Noetherian, but not all rings are Noetherian. For example,*k*[*X*1*, . . . , X**n**, . . .*]
is not Noetherian: *X*1*, . . . , X**n*is a minimal set of generators for the ideal (*X*1*, . . . , X**n*)
in *k*[*X*1*, . . . , X**n*], and *X*1*, . . . , X**n**, . . .* is a minimal set of generators for the ideal
(*X*1*, . . . , X**n**, . . .*) in *k*[*X*1*, . . . , X**n**, . . .*]

Proposition 1.2. *Every nonzero nonunit element of a Noetherian integral do-*
*main can be written as a product of irreducible elements.*

Proof. We shall need to use that

(*a*)*⊂*(*b*) *⇐⇒* *b|a*, with equality *⇐⇒* *b* =*a×*unit*.*

The ﬁrst assertion is obvious. For the second, note that if *a* = *bc* and *b* = *ad* then
*a*=*bc*=*adc*, and so *dc*= 1. Hence both *c* and *d* are units.

Suppose the statement is false, and choose an element *a* *∈* *A* which contradicts
the statement and is such that (*a*) is maximal among the ideals generated by such
elements (here we use that *A*is Noetherian). Since*a* can not be written as a product
of irreducible elements, it is not itself irreducible, and so*a* =*bc*with*b*and*c*nonunits.

Clearly (*b*)*⊃*(*a*), and the ideals can’t be equal for otherwise*c*would be a unit. From
the maximality of (*a*), we deduce that *b* can be written as a product of irreducible
elements, and similarly for *c*. Thus *a* is a product of irreducible elements, and we
have a contradiction.

4After Emmy Noether (1882–1935).

12 1. Preliminaries from Commutative Algebra

**Local rings.** A ring *A* is said to *local* if it has exactly one maximal idealm. In this
case, *A*^{×} =*A*m(complement of m in*A*).

Lemma 1.3 (Nakayama’s lemma). *Let* *A* *be a local Noetherian ring, and let* a *be*
*a proper ideal in* *A. Let* *M* *be a ﬁnitely generated* *A-module, and deﬁne*

a*M* =*{*

*a**i**m**i* *|a**i**∈*a*,* *m**i**∈M}.*
(a) *If* a*M* =*M, then* *M* = 0*.*

(b) *If* *N* *is a submodule of* *M* *such that* *N*+a*M* =*M, then* *N* =*M.*

Proof. (a) Suppose *M* = 0. Among the ﬁnite sets of generators for *M*, choose
one *{m*_{1}*, ..., m*_{k}*}* having the fewest elements. From the hypothesis, we know that we
can write

*m*_{k}=*a*_{1}*m*_{1}+*a*_{2}*m*_{2}+*..*.*a*_{k}*m*_{k} some *a*_{i} *∈*a*.*
Then

(1*−a*_{k})*m*_{k}=*a*_{1}*m*_{1}+*a*_{2}*m*_{2}+*..*. +*a*_{k}_{−}_{1}*m*_{k}_{−}_{1}*.*

As 1*−a*_{k}is not inm, it is a unit, and so*{m*_{1}*, ..., m*_{k}_{−}_{1}*}*generates *M*. This contradicts
our choice of*{m*_{1}*, ..., m*_{k}*}*, and so *M* = 0*.*

(b) We shall show thata(*M/N*) = *M/N*, and then apply the ﬁrst part of the lemma
to deduce that *M/N* = 0. Consider *m*+*N*, *m* *∈* *M*. From the assumption, we can
write

*m* =*n*+

*a*_{i}*m*_{i}, with *a*_{i} *∈*a, *m*_{i} *∈M.*

Whence

*m*+*N* =

*a*_{i}*m*_{i}+*N* =

*a*_{i}(*m*_{i}+*N*) (deﬁnition of the action of *A* on *M/N*)*,*
and so *m*+*N* *∈*a(*M/N*)*.*

The hypothesis that*M* be ﬁnitely generated in the lemma is crucial. For example,
if*A*is a local integral domain with maximal ideal m= 0, thenm*M* =*M* for any ﬁeld
*M* containing*A* but *M* = 0.

**Rings of fractions.** Let*A* be an integral domain; there is a ﬁeld*K* *⊃A*, called the
*ﬁeld of fractions* of*A*, with the property that every*c∈K* can be written in the form
*c*= *ab*^{−}^{1}, *a, b∈* *A*, *b* = 0. For example, Q is the ﬁeld of fractions of Z, and *k*(*X*) is
the ﬁeld of fractions of *k*[*X*]*.*

Let *A* be an integral domain with ﬁeld of fractions *K*. *A* subset *S* of *A* is said
to be *multiplicative* if 0 *∈/* *S*, 1 *∈* *S*, and *S* is closed under multiplication. If *S* is a
multiplicative subset, then we deﬁne

*S*^{−}^{1}*A*=*{a/b* *∈K* *|b* *∈S}.*
It is obviously a subring of*K.*

Example 1.4. (a) Let*t* be a nonzero element of*A*; then
*S*_{t}=^{df} *{*1,*t*,*t*^{2},...*}*

1. Preliminaries from Commutative Algebra 13

is a multiplicative subset of*A*, and we (sometimes) write*A*_{t} for*S*_{t}^{−}^{1}*A*. For example,
if *d* is a nonzero integer,

Z*d* =*{a/d*^{n} *∈*Q*|a∈*Z, *n≥*0*}.*

It consists of those elements of Qwhose denominator divides some power of *d*.

(b) If p is a prime ideal, then *S*_{p} =*A*p is a multiplicative set (if neither*a* nor *b*
belongs to p, then*ab* does not belong top). We write *A*_{p} for *S*_{p}^{−}^{1}*A*. For example,

Z(*p*) =*{m/n∈*Q*|n* is not divisible by *p}.*

Proposition 1.5. *LetAbe an integral domain, and letS* *be a multiplicative subset*
*of* *A. The map*

p*→S*^{−}^{1}p=^{df} *{a/s|a∈*p*,* *s∈S}*

*is a bijection from the set of prime ideals in* *Asuch that*p*∩S* =∅ *to the set of prime*
*ideals in* *S*^{−}^{1}*A; the inverse map is* q*→*q*∩A.*

Proof. It is easy to see that

p a prime ideal disjoint from *S* *⇒S*^{−}^{1}p is a prime ideal,

q a prime ideal in *S*^{−}^{1}*A⇒*q*∩A* is a prime ideal disjoint from *S,*
and so we only have to show that the two maps are inverse, i.e.,

(*S*^{−}^{1}p)*∩A*=pand *S*^{−}^{1}(q*∩A*) = q.

(*S*^{−}^{1}p)*∩A* =p: Clearly (*S*^{−}^{1}p)*∩A⊃*p. For the reverse inclusion, let*a/s∈*(*S*^{−}^{1}p)*∩A*,
*a* *∈*p, *s∈* *S*. Consider the equation ^{a}_{s} *·s*=*a* *∈* p. Both *a/s* and *s* are in *A*, and so
at least one of *a/s* or *s* is in p (because it is prime); but *s /∈* p(by assumption), and
so *a/s∈*p*.*

*S*^{−}^{1}(q*∩A*) = q : Clearly *S*^{−}^{1}(q*∩A*) *⊂* q because q*∩A* *⊂* q and q is an ideal in
*S*^{−}^{1}*A*. For the reverse inclusion, let*b∈*q. We can write it*b* =*a/s*with*a∈A*,*s* *∈S*.

Then *a*=*s·*(*a/s*)*∈*q*∩A*, and so *a/s*= (*s·*(*a/s*))*/s∈S*^{−}^{1}(q*∩A*)*.*

Example 1.6. (a) If p is a prime ideal in *A*, then *A*_{p} is a local ring (because p
contains every prime ideal disjoint from *S*_{p}).

(b) We list the prime ideals in some rings:

Z: (2)*,*(3)*,*(5)*,*(7)*,*(11)*, . . . ,*(0);

Z2: (3)*,*(5)*,*(7)*,*(11)*, . . . ,*(0);

Z(2): (2)*,*(0);

Z42: (5)*,*(11)*,*(13)*, . . . ,*(0);

Z*/*(42): (2)*,*(3)*,*(7).

Note that in general, for *t* a nonzero element of an integral domain,

*{*prime ideals of *A*_{t}*} ↔ {*prime ideals of*A* not containing *t}*
*{*prime ideals of *A/*(*t*)*} ↔ {*prime ideals of *A* containing *t}.*

14 1. Preliminaries from Commutative Algebra

**The Chinese remainder theorem.** Recall the classical form of the theorem: let
*d*1*, ..., d**n* be integers, relatively prime in pairs; then for any integers *x*1*, ..., x**n*, the
equations

*x≡x*_{i} (mod *d*_{i})

have a simultaneous solution *x∈*Z; if *x* is one solution, then the other solutions are
the integers of the form *x*+*md*, *m∈*Z, where*d*=

*d*_{i}*.*

We want to translate this in terms of ideals. Integers*m* and *n* are relatively prime
if and only if (*m, n*) = Z, i.e., if and only if (*m*) + (*n*) = Z. This suggests deﬁning
ideals a and b in a ring *A* to be *relatively prime* if a+b=*A*.

If*m*1*, ..., m**k*are integers, then*∩*(*m**i*) = (*m*) where*m*is the least common multiple
of the *m**i*. Thus *∩*(*m**i*) *⊃* (

*m**i*) =

(*m**i*). If the *m**i* are relatively prime in pairs,
then *m*=

*m**i*, and so we have *∩*(*m**i*) =

(*m**i*). Note that in general,
a1*·*a2*· · ·*a*n**⊂*a1*∩*a2*∩...∩*a*n**.*

These remarks suggest the following statement.

Theorem 1.7. *Let* a1*, ...,*a*n* *be ideals in a ring* *A, relatively prime in pairs. Then*
*for any elements* *x*_{1}*, ..., x*_{n} *of* *A, the equations*

*x≡x**i* (*mod*a*i*)

*have a simultaneous solution* *x∈A; if* *x* *is one solution, then the other solutions are*
*the elements of the form* *x*+*a* *with* *a* *∈ ∩*a*i**; moreover,* *∩*a*i* =

a*i**. In other words,*
*the natural maps give an exact sequence*

0*→*a*→A→* ^{n}

*i*=1

*A/*a*i* *→*0
*with* a=*∩*a*i* =

a*i**.*

Proof. Suppose ﬁrst that *n* = 2. Asa1 +a2 =*A*, there are elements*a*_{i}*∈*a*i* such
that *a*_{1}+*a*_{2} = 1. The element*x*=_{df} *a*_{1}*x*_{2}+*a*_{2}*x*_{1} has the required property.

For each *i* we can ﬁnd elements*a*_{i} *∈*a1 and *b*_{i} *∈*a*i* such that
*a*_{i}+*b*_{i} = 1, all*i≥*2*.*

The product

*i**≥*2(*a**i*+*b**i*) = 1, and lies in a1+

*i**≥*2a*i*, and so
a1+

*i**≥*2

a*i* =*A.*

We can now apply the theorem in the case *n* = 2 to obtain an element *y*_{1} of *A* such
that

*y*_{1} *≡*1 mod a1*,* *y*_{1} *≡*0 mod

*i**≥*2

a*i**.*

These conditions imply

*y*_{1} *≡*1 mod a1*,* *y*_{1} *≡*0 mod a*j*, all *j >*1*.*

Similarly, there exist elements*y*2*, ..., y**n*such that

*y**i* *≡*1 mod a*i**,* *y**i* *≡*0 mod a*j* for *j* =*i.*

The element*x*=

*x**i**y**i* now satisﬁes the requirements.

1. Preliminaries from Commutative Algebra 15

It remains to prove that*∩*a*i* =

a*i*. We have already noted that*∩*a*i* *⊃*

a*i*. First
suppose that *n* = 2, and let*a*1 +*a*2= 1, as before. For *c∈*a1 *∩*a2, we have

*c*=*a*_{1}*c*+*a*_{2}*c∈*a1*·*a2

which proves thata1*∩*a2 =a1a2. We complete the proof by induction. This allows us
to assume that

*i**≥*2a*i* =*∩**i**≥*2a*i*. We showed above that a1 and

*i**≥*2a*i* are relatively
prime, and so

a1*·*(

*i**≥*2

a*i*) =a1*∩*(

*i**≥*2

a*i*) =*∩*a*i**.*

The theorem extends to *A*-modules.

Theorem 1.8. *Let* a1*, ...,*a*n**be ideals in* *A, relatively prime in pairs, and let* *M* *be*
*an* *A-module. There is an exact sequence:*

0*→*a*M* *→M* *→*

*i*

*M/*a*i**M* *→*0
*with* a=

a*i* =*∩*a*i**.*

This has an elementary proof (see Janusz 1996, p. 9), but I prefer to use tensor products, which I now review.

**Review of tensor products.** Let*M*,*N*, and*P* be*A*-modules. A mapping*f*: *M×*
*N* *→P* is said to be *A-bilinear* if

*f*(*m*+*m*^{}*, n*) = *f*(*m, n*) +*f*(*m*^{}*, n*); *f*(*m, n*+*n*^{}) =*f*(*m, n*) +*f*(*m, n*^{})
*f*(*am, n*) = *af*(*m, n*) =*f*(*m, an*)*,* *a∈A,* *m, m*^{} *∈M,* *n, n*^{} *∈N,*
i.e., if it is linear in each variable. A pair (*Q, f*) consisting of an *A*-module *Q* and
an *A*-bilinear map *f* : *M* *×N* *→* *Q* is called the *tensor product* of *M* and *N* if
any other *A*-bilinear map *f*^{} : *M* *×N* *→* *P* factors uniquely into *f*^{} = *α◦f* with
*α* : *Q* *→* *P A*-linear. The tensor product exists, and is unique (up to a unique
isomorphism). We denote it by *M* *⊗**A* *N*, and we write (*m, n*) *→* *m⊗* *n* for *f*.

The pair (*M* *⊗**A**N,*(*m, n*) *→* *m⊗n*) is characterized by each of the following two
conditions:

(a) The map *M* *×* *N* *→* *M* *⊗**A* *N* is *A*-bilinear, and any other *A*-bilinear map
*M×N* *→P* is of the form (*m, n*)*→α*(*m⊗n*) for a unique*A*-linear map*α*: *M⊗**A**N* *→*
*P*; thus

Bilin*A*(*M×N, P*) = Hom*A*(*M* *⊗**A**N, P*)*.*

(b) As an *A*-module, *M* *⊗**A**N* generated by the symbols *m⊗n*, *m* *∈* *M*, *n* *∈* *N*,
which satisfy the relations

(*m*+*m*^{})*⊗n* = *m⊗n*+*m*^{}*⊗n*; *m⊗*(*n*+*n*^{}) =*m⊗n*+*m⊗n*^{}
*am⊗n* = *a*(*m⊗n*) =*m⊗an.*

Tensor products commute with direct sums: there is a canonical isomorphism
(*⊕**i**M*_{i})*⊗**A*(*⊕**j**N*_{j})*→ ⊕*^{≈} *i,j**M*_{i}*⊗**A**N*_{j}*,* (

*m*_{i})*⊗*(

*n*_{j})*→*

*m*_{i}*⊗n*_{j}*.*

16 1. Preliminaries from Commutative Algebra

It follows that if*M* and *N* are free *A*-modules^{5} with bases (*e**i*) and (*f**j*) respectively,
then *M* *⊗**A**N* is a free*A*-module with basis (*e**i**⊗f**j*). In particular, if*V* and *W* are
vector spaces over a ﬁeld *k* of dimensions *m* and *n* respectively, then *V* *⊗**k**W* is a
vector space over*k* of dimension*mn*.

Let*α*:*M* *→N* and *β*: *M*^{} *→N*^{} be *A*-linear maps. Then
(*m, n*)*→α*(*m*)*⊗β*(*n*) : *M* *×N* *→M*^{}*⊗**A**N*^{}

is *A*-bilinear, and therefore factors through *M* *×N* *→* *M* *⊗**A**N*. Thus there is an
*A*-linear map *α⊗β* :*M* *⊗**A**N* *→M*^{} *⊗**A**N*^{} such that

(*α⊗β*)(*m⊗n*) =*α*(*m*)*⊗β*(*n*)*.*

Remark 1.9. Let *α*: *k*^{m} *→* *k*^{m} and *β*: *k*^{n} *→* *k*^{n} be two matrices, regarded as a
linear maps. Then *α⊗β* is a linear map *k*^{mn} *→* *k*^{mn}. Its matrix with respect to
the canonical basis is called the *Kronecker product* of the two matrices. (Kronecker
products of matrices pre-date tensor products by about 70 years.)

Lemma 1.10. *If* *α*: *M* *→N* *and* *β*: *M*^{} *→N*^{} *are surjective, then so also is*
*α⊗β*: *M* *⊗**A**N* *→M*^{} *⊗**A**N*^{}*.*

Proof. Recall that *M*^{}*⊗N*^{} is generated as an *A*-module by the elements*m*^{}*⊗n*^{},
*m*^{} *∈* *M*^{}, *n*^{} *∈* *N*^{}. By assumption *m*^{} = *α*(*m*) for some *m* *∈* *M* and *n*^{} = *β*(*n*) for
some *n∈N*, and so*m*^{}*⊗n*^{} =*α*(*m*)*⊗β*(*n*) = (*α⊗β*)(*m⊗n*). Therefore Im(*α⊗β*)
contains a set of generators for *M*^{} *⊗**A**N*^{} and so it is equal to it.

One can also show that if

*M →M* *→M*^{} *→*0
is exact, then so also is

*M*^{}*⊗**A**P* *→M* *⊗**A**P* *→M*^{}*⊗**A**P* *→*0*.*

For example, if we tensor the exact sequence

0*→*a*→A* *→A/*a*→*0
with *M*, we obtain an exact sequence

a*⊗**A**M* *→M* *→*(*A/*a)*⊗**A**M* *→*0
The image of a*⊗M* in *M* is

a*M* =^{df} *{*

*a*_{i}*m*_{i} *|a*_{i} *∈*a, *m*_{i}*∈M},*
and so we obtain from the exact sequence that

*M/*a*M* *∼*= (*A/*a*A*)*⊗**A**M* (1.11).

By way of contrast, if *M* *→* *N* is injective, then *M* *⊗**A* *P* *→* *N* *⊗**A**P* need not
be injective. For example, take *A* = Z, and note that (Z *→*^{m} Z)*⊗*Z(Z*/m*Z) equals
Z*/m*Z*→*^{m} Z*/m*Z, which is the zero map.

5Let *M* b e an *A*-module. Elements *e*1*, . . . , e**m* form a *basis* for *M* if every element of *M* can
be expressed uniquely as a linear combination of the *e**i*’s with coeﬃcients in *A*. Then *A*^{m} *→* *M*,
(*a*1*, . . . , a**m*)*→*

*a**i**e**i**,* is an isomorphism of*A*-modules, and*M* is said to be a *free* *A-module of*
*rank* *m*.

1. Preliminaries from Commutative Algebra 17

Proof of Theorem 1.8. Return to the situation of the theorem. When we ten- sor the isomorphism

*A/*a*→*^{∼}^{=}
*A/*a*i*

with *M*, we get an isomorphism

*M/*a*M* *∼*= (*A/*a)*⊗**A**M* *→*^{∼}^{=}

(*A/*a*i*)*⊗**A**M* *∼*=

*M/*a*i**M,*
as required.

*Extension of scalars.* If*A→B* is an *A*-algebra and*M* is an *A*-module, then*B⊗**A**M*
has a natural structure of a *B*-module for which

*b*(*b*^{}*⊗m*) = *bb*^{}*⊗m,* *b, b*^{} *∈B,* *m∈M.*

We say that*B⊗**A**M* is the*B*-module obtained from *M* by*extension of scalars.* The
map *m* *→* 1*⊗m*: *M* *→* *B⊗**A**M* is uniquely determined by the following universal
property: it is*A*-linear, and for any*A*-linear map*α*: *M* *→N* from*M*into a*B*-module
*N*, there is a unique *B*-linear map *α*^{}: *B* *⊗**A**M* *→* *N* such that *α*^{}(1*⊗m*) = *α*(*m*).

Thus *α→α*^{} deﬁnes an isomorphism

Hom*A*(*M, N*)*→*Hom*B*(*B⊗**A**M, N*)*, N* a *B*-module)*.*

For example, *A* *⊗**A**M* = *M*. If *M* is a free *A*-module with basis *e*1*, . . . , e**m*, then
*B⊗**A**M* is a free *B*-module with basis 1*⊗e*1*, . . . ,*1*⊗e**m*.

*Tensor products of algebras.* If*f*: *A→B* and*g*: *A→C*are*A*-algebras, then*B⊗**A**C*
has a natural structure of an *A*-algebra: the product structure is determined by the
rule

(*b⊗c*)(*b*^{}*⊗c*^{}) =*bb*^{}*⊗cc*^{}
and the map*A* *→B* *⊗**A**C* is*a→f*(*a*)*⊗*1 = 1*⊗g*(*a*).

For example, there is a canonical isomorphism

*a⊗f* *→af* :*K* *⊗**k**k*[*X*1*, . . . , X**m*]*→K*[*X*1*, . . . , X**m*] (1*.*12)*.*

*Tensor products of ﬁelds.* We are now able to compute*K⊗**k*Ω if*K*is a ﬁnite separable
ﬁeld extension of *k* and Ω is an arbitrary ﬁeld extension of *k*. According to the
primitive element theorem (FT, 5.1), *K* = *k*[*α*] for some *α* *∈* *K*. Let *f*(*X*) be the
minimum polynomial of *α*. By deﬁnition this means that the map *g*(*X*) *→* *g*(*α*)
determines an isomorphism

*k*[*X*]*/*(*f*(*X*))*→K.*

Hence

*K⊗**k*Ω*∼*= (*k*[*X*]*/*(*f*(*X*)))*⊗*Ω*∼*= Ω[*X*]*/*(*f*(*X*))

by (1.11) and (1.12). Because *K* is separable over*k*,*f*(*X*) has distinct roots. There-
fore *f*(*X*) factors in Ω[*X*] into monic irreducible polynomials

*f*(*X*) =*f*_{1}(*X*)*· · ·f*_{r}(*X*)