and the number of elements in each class. Use it to show that the only normal subgroups of S

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2018 ALGEBRA MIDTERM EXAM

A COURSE BY CHIN-LUNG WANG AT NTU

1. Determine the conjugacy classes in S

5

and the number of elements in each class. Use it to show that the only normal subgroups of S

5

are 1, A

5

, S

5

. 2. Show that there are no simple groups of order 148 or of order 56.

3. Show that (i) a finite abelian group is cyclic ⇐⇒ exp G = | G | and (ii) any finite subgroup of the multiplicative group of a field is cyclic.

4. Let R be a ring. Prove that if a ∈ R has more than one right inverse then it has infinitely many. (Hint: if ab = 1, show first that 1 − ba 6= 0.) Give an example such that this fails for monoids.

5. Let I be an ideal in a ring R. Show that M

n

( I ) is an ideal in M

n

( R ) and every ideal in M

n

( R ) is in this form.

6. Let I

1

, . . . , I

n

be pairwise coprime (two-sided) ideals in a ring R in the sense that I

i

+ I

j

= R for i 6= j. Show that (i) R/ (

Tni=1

I

i

) ∼ =

ni=1

R/I

i

and (ii) if moreover that R is commutative then

Tni=1

I

i

=

ni=1

I

i

.

7. Let F be a field with | F | = q. Show that (i) the ring of polynomial functions in r variables over F is isomorphic to F [ x

1

, . . . , x

r

] / h x

q1

− x

1

, . . . , x

qr

− x

r

i and (ii) every function f : F

r

→ F is a polynomial function.

8. Let D be a Euclidean domain whose function δ satisfies: (i) δ ( ab ) = δ ( a ) δ

(

b ) and (ii) δ ( a + b ) ≤ max ( δ ( a ) , δ ( b )) . Show that either D is a field or D = F [ x ] where F is a field.

* You may prove the existence of Sylow p-subgroups to supplement one (and only one) problem up to 10 pts. Indicate explicitly the problem number.

** Bonus: Let A = Z [

12

( 1 + √

− 19 )] . Show that (i) the norm N : A → N is a Dedekind–Hasse norm and (ii) A is not a Euclidean domain.

Date: Time and place: pm 1:20 – 5:20, November 9, 2018 at AMB 101.

Note: (1) each problem is of 15 points (total 120 pts), (2) you may work on each part separately, (3) show your answers/computations/proofs in details.

1

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