Math 594. Solutions 2
Book problems 4.1:
9. Suppose that G acts transitively on the nite set A and let H be a normal subgroup of G with O1 . . . Or
the distinct orbits of H on A.
(a) Prove that G permutes the sets O1 . . . Or transitively. Deduce that all
Math 594. Solutions 1
1. Let V and W be nite-dimensional vector spaces over a eld F . Let G = GL(V ) and H = GL(W ) be
the associated general linear groups. Let X denote the vector space HomF (V, W ) of linear maps from V to
W , but viewed only as a set (
Math 594. Solutions to final exam
1. pts) Work out the Galois group of X 4 7 over each of the following elds: Q, Q( 7), Q( 7),
(20
Q( 1). Determine the lattice of subelds for the case of Q as the base eld.
Solution By Eisensteins criterion, f = X 4 7 is i
Math 594. Solutions to Exam 1
1. (20 pts) Let G be a group. We dene its automorphism group Aut(G) to be the set of group isomorphisms
: G G.
(i) (5 pts) Prove that using composition of maps, Aut(G) is a group.
(ii) (5 pts) For g G, dene cg : G G to be th
Solutions to Homework 12
1. (i) Let k be a nite eld, with k /k a nite extension with degree d. Prove that Gal(k /k ) is a cyclic
group of order d, with x x|k| a generator (called the Frobenius map).
(ii) What is the size of a splitting eld of X 15 2 over
Solutions to Homework 11
n
1. Let k be a nite eld with size q . Show that in k [T ], T q T factors into the product of all monic
irreducible polynomials of degree dividing n, each appearing exactly once in the factorization.
n
Since T q T is separable ove
Solutions to Homework 10
1. Let L1 /L2 /k be a tower of algebraic extensions, with Li /k normal. Prove that there is a natural surjection
of groups Aut(L1 /k ) Aut(L2 /k ), with kernel Aut(L1 /L2 ).
Using the given tower, we regard k and L2 as subelds of
Solutions to Homework 9
1. Read 2 in Appendix 2 in Langs Algebra (3rd edition). Find the place in his proof of Zorns Lemma (pp.
881-884) where the axiom of choice is used. Write down where it is.
In the proof of Cor 2.4, he constructs f : S S . This requi
Solutions to Homework 8
1. Let L1 and L2 be two nite extensions of k inside of an extension L/k .
(i) Prove [L1 L2 : k ] [L1 : k ][L2 : k ].
(ii) Assume that [L1 : k ] and [L2 : k ] are relatively prime. Show that [L1 L2 : k ] = [L1 : k ][L2 : k ]. If we
Solutions to Homework 7
1. For a eld k , show that the set Aut(k ) of automorphisms of k forms of group under composition. If
f Z[X ] and a k satises f (a) = 0, then show that for any Aut(k ), f ( (a) = 0. If L is an extension
of k and Aut(L) acts trivial
Math 594. Solutions to Homework 6
1. Let R be a ring. Prove that for all x R, 0R x = 0R and (1R )x = x.
Since 0R + 0R = 0R , if we multiply both sides by x and add (0R x) to both sides and use the associative
law for addition, we get 0R x = 0R . Since 1R
Math 594. Solutions 5
Book problems 6.1:
7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for
innite groups). Give an example of a group G which possesses a normal subgroup H such that H and G/H
are nil
Math 594. Solutions 4
Book problems 4.5:
6. Exhibit all Sylow 3-subgroups of S4 and A4 .
Solution: Any Sylow 3-subgroup of S4 or A4 has size 3 and is therefore generated by an element of
order 3. Hence, the Sylow 3-subgroups are specied by elements of ord
Math 594. Solutions 3
Book problems 5.1:
14. Let G = A1 A2 An and for each i let Bi
Ai . Prove that B1 B2 Bn
(A1 A2 An )/(B1 B2 Bn )
G and that
(A1 /B1 ) (A2 /B2 ) (An /Bn ).
Solution: Dene
: G (A1 /B1 ) (A2 /B2 ) (An /Bn )
by (a1 , a2 , . . . , an ) (a1