SOLUTIONS TO PROBLEMS V
1.(i) This is essentially the same as an argument seen in class. Let E denote a
splitting eld for xp t over J. Write h(x) = xp t. Since E is a splitting eld
for h, there exists some E with h() = 0. In particular, one has p = t.
But
Galois Theory: Exercise set 3 Sample Solutions
Due in class on Monday, 10 November 2014
1. Suppose that L : K is a eld extension with K L, and that Gal(L : K).
Suppose that g K[t] and that g is irreducible over K. Show that for L,
g() = 0 if and only if g
SOLUTIONS TO PROBLEMS IV
1. Let L : K be a nite normal extension, and suppose that f is irreducible
over K[x]. Let M : K be a splitting eld extension for f over K, so that M : K
is normal by Theorem I.32. We put L = LM , and note from Theorem II.2(ii)
tha
Galois Theory: Exercise set 2
Due in class on Monday, 27 October 2014
1. Suppose K, L are elds so that Q K, Q L, and : K L is a homomorphism.
(a) For b K , show that b1 = (b)1 .
(b) Show that leaves Q pointwise xed. (Suggestion: Use that (1) = 1.)
Solutio
Galois Theory: Exercise set 1
Due in class on Monday, 14 October 2014
1. Let K be a eld; recall that the polynomial ring K[t] is a unique factorisation
domain. Recall also that a non-zero polynomial f K[t] is monic if its leading
coecient is 1, meaning f
GALOIS THEORY: REVIEW OF WEEKS 1-6
Unless otherwise stated, the denitions, lemmas, propositions, theorems
and corollaries are in the Galois Theory Lecture Notes by Andrew Booker
and Abhishek Saha, which were distributed in class and are posted to my
websi
GALOIS THEORY: PROBLEMS IV
TO BE HANDED IN BY MONDAY 24TH NOVEMBER 2014
1. Suppose that L : K is a nite normal extension and that f is an irreducible
polynomial in K[x]. Suppose that g and h are irreducible monic factors of f
in L[x]. Show that there is a
GALOIS THEORY: PROBLEMS V
TO BE HANDED IN BY MONDAY 8TH DECEMBER 2014
1. Let p be a prime number, let Fp denote the nite eld of p elements, and let
K = Fp (t). Suppose that L : K is a eld extension, and s L is transcendental
over K. Write J = K(s), and le
Galois Theory: Exercise set 3
Due in class on Monday, 10 November 2014
1. Suppose that L : K is a eld extension with K L, and that Gal(L : K),
f K[t]. Suppose that g K[t] is a factor of f in K[t], and that g is irreducible
over K. Show that for L, g() = 0
Galois Theory: Exercise set 2
Due in class on Monday, 27 October 2014
1. Suppose K, L are elds so that Q K, Q L, and : K L is a homomorphism.
(a) For b K , show that b1 = (b)1 .
(b) Show that leaves Q pointwise xed. (Suggestion: Use that (1) = 1.)
2. Supp
Galois Theory: Exercise set 1
Due in class on Monday, 14 October 2014
1. Let K be a eld; recall that the polynomial ring K[t] is a unique factorisation
domain. Recall also that a non-zero polynomial f K[t] is monic if its leading
coecient is 1, meaning f