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
56
7
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
62
9
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
31
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(),
MA241 Combinatorics Marking Sheet 3
Deadline: Wednesday, 2 March 2005, 2:00.
For this sheet (B1), (B2), (B5), (B7)(cdef) are marked.
(B1) Let S denote the set of fundamental reections in 7 and write si < sj
if and only if i < j. Find the lexicographically
MA241 Combinatorics Marking Sheet 1
Deadline: Wednesday, 26 January 2005, 2:00.
For this sheet, B1(b), B2, B6(abc) and B7 will be assessed.
Marks are in the margin. (Each assessment has 25 points in total.)
Correct answers always get full marks even if th
MA4F2 Braid Groups Sheet 4
Deadline: Friday, 11 March 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
(A1) Let G, H be a groups and let
H G H
(x, g) xg
den
MA241 Combinatorics Marking Sheet 2
Deadline: Wednesday, 9 February 2005, 2:00.
For this sheet, (B2)cdef, (B4), (B5), (B6) will be assessed.
+
(B2) Let B3 be the monoid presented by (S1 , R1 ) = (1, 2 | 121 = 212).
(Warning: is the identity, 1 is not.) Le
THE UNIVERSITY OF WARWICK
FOURTH YEAR EXAMINATION: never
MOCK EXAM MA4F20 BRAID GROUPS 20042005
Time Allowed: 3 hours
Read carefully the instructions on the answer book and make sure that the particulars
required are entered on each answer book.
Calculato
MA4F2 Braid Groups Sheet 3
Deadline: Wednesday, 2 March 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
(A1) Let x = si and y = sj .
(a) Suppose |i j| = 1.
MA4F2 Braid Groups Sheet 1
Deadline: Wednesday, 26 January 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
(A1) Prove that the following spaces are path-co
March 8, 2005
MA4F2 Braid Groups
21
(c) Deduce a contradiction.
(d) Finish the proof.
6.10 Rewriting systems. Let (S, R) be a monoid presentation. The biinvariant closure of R is
:= (axb, ayb) (x, y) R, a, b S .
R
We call (S, R) a rewriting system if is
March 8, 2005
MA4F2 Braid Groups
59
Figure 15:
j
i
(a): aij
(b):
16.13 Exercise. Suppose j < k < < i. We made a relation (16.12) as a
consequence of the relations (16.4) and (16.5) and whose left hand side starts
with aik and its right hand side with aj
March 8, 2005
MA4F2 Braid Groups
40
(gy)(gx)(gy). In a similar way, prove yourself that (gsi )(gsj ) = (gsj )(gsi ) if
|i j| > 1. This proves that v is well-dened.
It is clear that uv = 1, so that in particular v is injective. It remains
to show that v is
MA4F2 Braid Groups Sheet 2
Deadline: Wednesday, 9 February 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
Needless to say, you may use previous parts of q
March 8, 2005
1
MA4F2 Braid Groups
2
Vague denition of braid groups
Figure 1: A braid on 4 strings
This is up
1.1 This section is purposely vague. Later well do things more precisely.
1.2 Vague denition of braids. Fix n 1. A braid on n strings, or an
n-br
MA241 Combinatorics Sheet 3
Deadline: Monday, 13 November 2006, 3:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA241 Combinatorics box in front of the Undergraduate Oce.
(A1) In the lectures we unfolded the basic recur
MA241 Combinatorics Sheet 2
Deadline: Monday, 30 October 2006, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA241 Combinatorics box in front of the Undergraduate Oce.
(B1) Let n 0. Express Tn =
5
k
in terms of n and
MA241 Combinatorics Sheet 5
Deadline: Monday, 15 January 2007, 3:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA241 Combinatorics box in front of the Undergraduate Oce.
m
m+1
Bj = [m = 0].
j
(A1) Use binomial convoluti
MA241 Combinatorics Marking Sheet 5
Deadline: Monday, 15 January 2007, 3:00.
For this sheet, B3(a) and B6 will be assessed.
(B3).
(a) Let t R. Find a closed formula for the exponential generating function
for the numbers an dened by
n
a0 = 1,
an+1 = t
k=0
MA241 Combinatorics Marking Sheet 1
Deadline: Monday, 23 October 2006, 2:00.
For this sheet, (B4)(b) and (B10) will be assessed.
(B4)(b). Use the summation factor method to solve the following, that is,
to nd a closed formula for bn (n 0).
(b) b0 = 0, bn