612 Example Sheet 4
Paul Hacking
28 March 2011
(1) Let k be an innite eld. Let f k[x1 , . . . , xn ] be a nonzero polynomial. Show that there exist a1 , . . . , an k such that f (a1 , . . . , an ) = 0.
[Hint: Use k[x1 , . . . , xn ] = k[x1 , . . . , xn1 ]
612 Example Sheet 3
Paul Hacking
28 February 2011
Notation: For F a eld and f F [x] a separable polynomial, the Galois
group of f over F is the Galois group G = Aut(K/F ) of the splitting eld
K of f over F .
(1) Let K/F be a Galois extension with group G
612 Example Sheet 5
Paul Hacking
11 April 2011
(1) Let A Q be a subring. Show that A = S 1 Z, where S Z is the
multiplicative subset generated by a set of primes T = cfw_pi | i I Z.
(Note that T is not necessarily nite.)
(2) Let A be a ring and p A a prim
612 Example Sheet 1
Paul Hacking
30 January 2011
Notation: For K/F a eld extension and K an element that is algebraic
over F , the degree of over F is the degree of the minimal polynomial of
over F , or, equivalently, the degree [Q() : Q] of the eld exte
612 Example Sheet 6
Paul Hacking
27 April 2011
Unless otherwise stated all groups G are nite and all representations
are nite dimensional complex representations. We say that a representation
: G GL(V ) of a group G is faithful if is injective. These pr
612 Example Sheet 2
Paul Hacking
15 February 2011
(1) For each of the following polynomials, describe the splitting eld K
over Q and nd the degree [K : Q].
(a) x4 2.
(b) x6 4.
(c) x4 + 2. [Hint: cf. HW1 Q9(b)]
(d) x4 + x2 + 1.
(2) (a) Let K/F be a splitti
612 Final Exam
Paul Hacking
2 May 2011
This is a take home exam, due Monday May 9 at 5PM in my mailbox.
You are allowed to consult your notes and textbooks. Please do not discuss
the exam with other students. There are 5 questions worth 10 points each.
Sh
ALGEBRA 611, FALL 2009. HOMEWORK 1
This homework is due before the class on Monday September 21. These
problems will be discussed during the review section on Monday at 4pm.
The grader will grade 5 random problems from this assignment. A problem with mult
ALGEBRA 611, FALL 2009. TAKE-HOME MIDTERM.
The take-home midterm is due before class on Monday Nov 23. You can
use lecture notes and the textbook. However, please do not discuss these
problems with each other or use internet or other textbooks. I will ask
ALGEBRA 611, FALL 2009. HOMEWORK 5
(1)
In this worksheet p denotes a prime number, k denotes an arbitrary eld,
Fp = Z/pZ denotes a nite eld with p elements, R denotes a ring, and R
denotes the multiplicative group of units of R.
1. Let R = Z[i] C. (a) Dra
ALGEBRA 611, FALL 2009. HOMEWORK 4
This homework is due before the class on Monday October 26. These
problems will be discussed during the review section on Monday at 4pm.
The grader will grade 5 random problems from this assignment. A problem with multip
ALGEBRA 611, FALL 2009. HOMEWORK 2
This homework is due before the class on Monday September 28. These
problems will be discussed during the review section on Monday at 4pm.
The grader will grade 5 random problems from this assignment. A problem with mult