11 May 2015
14.5.1 Determine the minimal polynomials satised by the primitive generators given in the text for
the subelds of Q(13 ).
Proof. The minimal polynomials satised by the primitive generators are:
+ 1 : x6
18 May 2015
1. Let K/Q be a Galois extension with Galois group G. Prove there exists a unique maximal
subeld F K such that:
(a) F/Q is Galois with abelian Galois group.
Proof. Let G = [G, G] and let F be the xed eld
6 April 2015
13.1.1 Show that p(x) = x3 + 9x + 6 is irreducible in Q[x]. Let be a root of p(x).
Find the inverse of 1 + in Q().
Proof. By Eisensteins criterion and Gauss lemma, since 3 divides 9 but 32
does not divi
18 May 2015
1. Find, explicitly, the proper, nontrivial subgroups of the following groups:
(a) G = (Z/16Z) .
Proof. Since G = cfw_1, 3, 5, 7, 9, 11, 13, 15, |G| = 8. If H G has order 2, then it consists
of the ide
3 June 2015
1. The subelds of Q(17 ) correspond to the subgroups of (Z/17Z) Z/16Z. A generator for
this cyclic group is the automorphism = 3 which maps = 17 to 3 . The nontrivial
subgroups correspond to the nontri
20 April 2015
1. (a) Show that if the eld K is generated over F by the elements 1 , . . . , n
then an automorphism of K xing F is uniquely determined by (1 ), . . . , (n ).
In particular show that an automorphism xe
12 April 2015
13.4.1 Determine the splitting eld and its degree over Q for p(x) = x4 2.
Proof. Call the splitting eld K. The roots of p(x) are 4 2, i 4 2. Since
K contains4 2 and i 4 2, it contains their quotient i.