Math 581 Assignment 8 Solutions
Recall that K/F is a normal extension if K is the splitting eld over F of some polynomial
in F [x]. This terminology is used in Problem 2.
1. Let K/F be a eld extension and let f (x), g (x) F [x]. Prove that the gcd of f (x
Math 581 Assignment 5 Solutions
1. Let R be a ring and set Z (R) = cfw_a R : ax = xa for all x R.
(a) Prove that Z (R) is a subring of R.
(b) View R as a subring of Mn (R) by thinking of r R as the diagonal matrix all of
whose entry is r, prove that Z (Mn
Math 581 Assignment 4 Solutions
1. Let G be a group and N a subgroup of N . Prove that there is a bijection between the
set of right cosets of N in G and the set of left cosets of N in G.
Solution. As stated, one way to prove this is to use the identity f
Math 581 Assignment 3
1. Let G be a group. If a, b G with ab = ba, prove that (ab)n = an bn for each n N.
Solution. We rst prove that abn = bn a, and we use induction on n. The case n = 1
is obvious. Suppose the result holds for n. Then abn+1 =
Math 581 Assignment 2
due Friday 7 September
1. Let : G H be a homomorphism.
(a) Prove that (eG ) = eH and, for each a G, that (a1 ) = (a)1 .
(b) If is surjective and G is Abelian, prove that H is Abelian.
Solution. (a) We have (eG ) = (eG eG ) = (eG )(eG
Math 581 Assignment 6 Solutions
1. Let R be a commutative ring and let I be an ideal of R. By Problem 3 of Assignment
5, from the ring homomorphism R (R/I )[x] that is the composition of the homomorphisms R R/I (R/I )[x] we get a ring homomorphism : R[x]
Math 581 Assignment 7 Solutions
1. Suppose K/F is an algebraic eld extension, and let R be a subring of K with F R.
Prove that R is a eld.
(The hypothesis that K/F is algebraic is necessary; consider F F [x] F (x).)
Solution. To prove that R is a eld we o
Math 581 Assignment 11 Solutions
1. Let q be a power of a prime p and let n be a positive integer with gcd(n, q ) = 1. If K
is the splitting eld of xn 1 over Fq , prove that |K | = q r , where r is the order of q in
(Zn ) .
(Hint: when does Fpn Fpm ?)
Math 581 Assignment 10 Solutions
1. Determine all the intermediate elds of K/Q, where K is the splitting eld over Q of
x3 2. Give some reason why you know you found all of them.
Solution. We have seen that if G = Gal(K/F ), then G S3 . Write G = , with
Math 581 Assignment 9 Solutions
1. Let K = Q( 3 2, ), where = e2i/3 . Determine Gal(K/Q).
Solution. The minimal polynomials of 3 2 and are x3 2 and x2 + x +1, respectively,
as we have seen in class. If Gal(K/Q), then must send 3 2 to a root of x3 2.
Math 581 Assignment 1
1. Let S be a nite set with an associative binary operation such that, for each x, y, z
S , we have
(a) If x y = x z , then y = z ,
(b) If y x = z x, then y = z .
Prove that (S, ) is a group. Furthermore, nd an exampl