MA3D5 Galois Theory Sheet 1
Deadline: Monday, 26 January 2009, 3:00.
Please put your solutions into the MA3D5 Galois Theory box in
front of the Undergraduate Oce. Mention your department if it is
not mathematics.
(1.1) Let , , be the roots of the equation
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March 9, 2009
Finite elds
If p is a prime number, we write F p := Z/( p). Warning: later we shall dene
Fq for more values of q, but in these cases it is not Z/(q).
Let K be a nite eld. Then its characteristic is a prime number p because otherwise K w
Test MA3D5 Galois Theory
Version 1 Monday 18 February 2007
Name:
Student Number:
This test covers the material lectured up to and including section 3.1. You can bring any other
notes you like, but no calculators.
Each question is worth 1 point. A correct
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March 9, 2009
Radical extensions
Keywords: Normal closure, solvable group, commutator, radical extension,
solvable extension.
9.1
Normal closures
Denition 116. Let K L M be elds with L/K nite. We say that M is a
normal closure of L/K if:
The eld M i
MA3D5 Galois Theory
13
2 Background on rings and elds
Keywords: Field of fractions, rational function, ideal, generators of an
ideal, kernel, coset, prime ideal, maximal ideal, quotient ring, principal ideal,
PID, UFD, rst isomorphism theorem for rings, c
MA3D5 Galois Theory
4
4.1
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Foundations of Galois theory
Closure correspondences
In this subsection, we x two disjoint sets A, B and a subset R A B, often
known as a binary relation. For all X A and Y B we dene
X := cfw_b B | (a, b) R for all a X ,
Y :=
MA3D5 Galois Theory
5
43
Normal subgroups and stability
Keywords: Algebraic extensions; nite extensions; nitely generated; normal subgroup; stable intermediate eld.
5.1
Algebraic eld extensions
Denition 84. A eld extension K L is said to be algebraic if e
MA3D5 Galois Theory Sheet 5
Deadline: Thursday 7 May 2009, 3:00.
Please put your solutions into the MA3D5 Galois Theory box in
front of the Undergraduate Oce. Mention your department if it is
not mathematics.
(5.1) Let K L be nite elds. Prove that L is se
MA3D5 Galois Theory Sheet 2
Deadline: Thursday, 5 February 2008, 3:00.
Question (2.5) is not for handing in. Please put your solutions into
the MA3D5 Galois Theory box in front of the Undergraduate Oce.
Mention your department if it is not mathematics.
(2
MA3D5 Galois Theory Sheet 3
Deadline: Thursday, 26 February 2008, 3:00.
Question (3.6) is not for handing in. Please put your solutions into
the MA3D5 Galois Theory box in front of the Undergraduate Oce.
Mention your department if it is not mathematics.
(
MA3D5 Galois Theory Sheet 4
Deadline: Monday, 9 March 2009, 3:00.
Please put your solutions into the MA3D5 Galois Theory box in
front of the Undergraduate Oce. Mention your department if it is
not mathematics.
(4.1) Let f = X 6 + 3, C, f () = 0, K = Q(),