Exercise 1 Please hand in (to me in Tuesdays class) correct answers to all the problems you got
wrong in the April 8 exam.
1. Let L/K be a Galois extension with Galois group G = Gal(L/K). Let H G
be a subgroup of G. Let
1. Consider the quotient ring Z[x]/(30x 1). Then 15x = 21 because 15x 2 = 1.
10x = 31 and 6x = 51 .
2. Let A be a ring and let S A. Suppose we have two universal solutions,
H1 : A cfw_S 1 A1 and H2 : A cfw_S 1 A2 , there exists i : S 1 . There
HW Math 123
Homework 4: Math 123,
Problem. Show that every element in S U2 is conjugate (in S U2 ) to a diagonal matrix.
Applying the spectral theorem for normal operators, we see that every element S S U2 is conjugate by
a unitary matrix U to a diagonal
More Galois Theory
HOMEWORK DUE APRIL 8:
The full primitive Element Theorem
These exercises in this subsection prove
Theorem 1 Let L/k be a separable eld extension nite degree. Then there is some L such
that L = k[alpha].
Exercise 1 S
For basic information about extension elds and degrees: Read
[Artin] pp. 493-499.
[Cox] pp. 88-99, and/or
For the construction of splitting elds:
[Artin] pp. 506-508, and/or
[Cox] pp. 55-62
[Lang] pp. 236-238.
Math 123 Homework 6
P ROBLEM 1.
Throughout this problem, let k := Fp (s) and f (t) = tp s k[t].
(a) We can view f = tp s as an element of Fp [s][t] whose coefcients are polynomials in s. It is clear that
s does not divide 1 1 (s).
s divides 0 and s itse
Math 123: Galois Theory
[Ex.1] Determine the irreducible polynomial for i +
The polynomial that splits i +
2 over Q.
2 also splits all its conjugates roots, so the desired polynomial is:
(x i 2)(x i + 2)(x + i 2)(x + i + 2)
= (x i)2 2)(x + i)2 2)
= (x2 3
Problem Set 8
Exercise 1. Show that it suffices to prove the theorem for separable field extensions L/k of finite degree such
that L is generated by two elements over k.
Solution. This is because once we prove the theorem for L generated by two elements,
Math 123 Homework 9
Suppose H is normal in G. This means H 1 for all G so that G(L/M ) = G(L/M ). This implies that
M = M for all . Thus every K-automorphism of L carries M to itself and hence denes a K-automorphism of M
by restriction. This re
An example of a universal Problem and it universal solution
The Universal solution to the problem of inverting a given
set of elements in a ring.
Exercise 1 The set of all rational numbers of the form a/2n for some integer a and some natural
number n s
Abelian representations, and character groups
Let G be a group (any group). Let K be a eld. A one-dimensional representation r of the group
G over the eld K can be given in various equivalent ways:
1. by giving V a one-di