REVIEW SHEET FOR LECTURE 9/27 AND 9/29
MING-LUN HSIEH
1. The ring of p-adic integers Zp: analytic properties For more details, you may look at [Ser73, Chapter II].
Definition 1.1. Define
Zp :=
(an)∈ ∏
n∈Z+
Z/pnZ | an+1 ≡ an(mod pn) for all n∈ Z+
.
We call the Zp the set of p-adic integers.
Proposition 1.2. The set Zp is a commutative ring with the obvious addition and multipli- cation, and Z can be embedded in Zp as a subring by m 7→ (m, m, · · · , m, · · · ).
The map Zp → Z/pnZ given by a = (an)7→ an is called the reduction modulo pn. We will write an≡ a (mod pnZp).
For every p-adic integer x = (a1, a2,· · · , an,· · · ) ∈ Zp with an ∈ Z/pnZ, there exist a sequence of numbers x1, x2,· · · , xn· · · ∈ {0, 1, · · · , p − 1} such that
a1 = x1(mod p)
a2 = x1+ x2p (mod p2)
a3 = x1+ x2p + x3p2(mod p3) ... ...
an+1 = x1+ x2p + x3p2+· · · xnpn = an+ xnpn(mod pn+1).
It is convenient to write the p-adic integer x uniquely as
x =
∑∞ n=0
xnpn, 0≤ xn≤ p − 1.
The above expansion is called the p-adic expansion of x.
Example 1.3. The polynomial x2+ 1 has a root in Z5. The ring Zp has a filtration:
Zp ⊃ pZp ⊃ p2Zp ⊃ · · · ⊃ pnZp ⊃ · · · . For r ∈ Z+, we can verify that
a = (an)∈ prZp ⇐⇒ ar≡ 0 (mod prZ).
Date: December 13, 2011.
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Definition 1.4 (Absolute value on Zp). We define the p-adic valuation vp : Zp → Z ∪ {∞}
as follows. We define vp(0) =∞. For every nonzero a ∈ Zp, we let vp(a)∈ Z≥0 be the unique integer such that
a∈ pvp(a)Zp but a ̸∈ pvp(a)+1Zp. We define the absolute value |·| : Zp → R≥0 by
|a| := p−vp(a)∈ R≥0. It is obvious that
|a| < 1 ⇐⇒ a ∈ pZp ⇐⇒ a ≡ 0 (mod pZp).
Lemma 1.5. For every a, b∈ Zp, we have (1) |a| = 0 ⇐⇒ a = 0.
(2) vp(ab) = vp(a) + vp(b) and |ab| = |a| |b|.
(3) |a + b| ≤ max {|a| , |b|}.
(4) |a + 1| = 1 if |a| < 1, or equivalently vp(a + b) = vp(a) if vp(a) < vp(b).
In view of the above lemma (3), |·| defines a distance function on Zp, and hence we can consider the topology on Zp induced by |·|. Every a = (a1, a2,· · · , an,· · · , ) ∈ Zp with an+1 ≡ an(mod pn). Then we have
nlim→∞|an− a| = 0.
Therefore, we can write limn→∞an = a. This shows that Z⊂ Zp is dense.
Proposition 1.6. Given a sequence {an}n=1,2,··· ∈ Zp such that limn→∞an = 0, then the series
∑∞ n=1
an converges in Zp.
Example 1.7. We define exponential function exp : Zp → Zp by exp(x) := 1 + x +x2
2 · · · =
∑∞ n=0
xn n. Similarly, we can define log : 1 + 2pZp → Zp by
log(1 + x) =
∑∞ n=1
(−1)n+1· xn n . Theorem 1.8. The topological space Zp is compact and complete.
The following lemma is a fundamental tool for solving equations in p-adic numbers.
Lemma 1.9 (Hensel’s lemma). Let f (x)∈ Zp[x]. If there exists x1 ∈ Zp such that
|f(x1)| < 1 and |f′(x1)| = 1, then there exists a unique a∈ Zp such that
f (a) = 0, |a − x1| < 1.
REVIEW SHEET FOR LECTURE 9/27 AND 9/29 3
2. The ring of p-adic integers Zp: algebraic properties We study the algebraic nature of Zp.
Proposition 2.1. The ring Zp is a principal ideal domain (PID).
Definition 2.2. The p-adic field Qp is the quotient field of Zp. It is clear that Qp ⊃ Q and Qp =∪∞n=0p−nZp. In addition, we have
Q×p = pZ· Z×p.
The absolute value|·| on Zp is extended to Qp so that|pnb| = p−nfor every n∈ Z and b ∈ Z×p. Theorem 2.3. The only automorphism of Qp is the identity automorphism.
Let Z×p be the multiplicative group of units in Zp. Proposition 2.4. We have
Z×p =Zp− pZp
={a ∈ Zp | a ̸≡ 0 (mod p)}
={a ∈ Zp | |a| = 1} . For every n∈ Z≥0, we put
Un :={
a∈ Z×p | a ≡ 1 (mod pn)}
= 1 + pnZp. By definition, U0 = Z×p. Then{Un} gives a filtration of Z×p:
Z×p = U0 ⊃ U1 ⊃ U2 ⊃ · · · ⊃ Un ⊃ · · · . Proposition 2.5. For n∈ Z+, we have a group isomorphism:
Un/Un+1 ≃ Fp. When n = 0, we have U0/U1 ≃ F×p.
Theorem 2.6. If p̸= 2, then we have a decomposition
Q×p = pZ· µp−1· (1 + pZp)≃ Z×Z/(p − 1)Z×Zp. References
[Ser73] J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973, Translated from the French, Graduate Texts in Mathematics, No. 7.
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3. Exercises
Exercise 3.1. Write down the 5-adic expansion of 15,−2, −3.
Exercise 3.2. Let p be an odd prime. Let a∈ Z with p - a. Show that (1) x2− a = 0 has a solution in Zp ⇐⇒ (
a p
)
= +1.
(2) x2 = p has no solution in Zp. (3) xp−1 = 1 has p− 1 solutions in Zp.
Exercise 3.3 (Strong form of Hensel’s lemma). Let f (x)∈ Zp[x]. Let a0 ∈ Zp such that
|f(a0)| < |f′(a0)|2, Then there is a solution of
f (a) = 0 such that |a − a0| ≤ |f(a)|
|f′(a)|.
Here f′(x) is the usual derivative of a polynomial f (x), and |·| is the p-adic absolute value.
Exercise 3.4. Let p̸= ℓ be two distinct primes. Show that the only (additive) group homo- morphism f : Zp → Zℓ is the zero map. In addition, find a necessary condition between p and ℓ under which the only (multiplicative) group homomorphisms f : Z×p → Z×ℓ is the trivial map.
Department of Mathematics, National Taiwan University, Taipei, Taiwan E-mail address: mlhsieh@math.ntu.edu.tw
