2018 ALGEBRA MIDTERM EXAM
A COURSE BY CHIN-LUNG WANG AT NTU
1. Determine the conjugacy classes in S
5and the number of elements in each class. Use it to show that the only normal subgroups of S
5are 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
nbe 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=1I
i) ∼ = ∏
ni=1R/I
iand (ii) if moreover that R is commutative then
Tni=1I
i= ∏
ni=1I
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
ri 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.
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