MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 7
Rik Sengupta November 20, 2009
1. Let n = pm1 . . . pmt be the prime factorization. Prove that t 1 n (x) = p1 .pt (xp1
m1 -1 m -1 .pt t
).
Solution. We will first show that if p|k,
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #2
Week #1: September 28th - October 4th. Topics: Universal property of quotients, the isomorphism theorems, subgroups of quotients, direct products, structure of finitely generated abelian groups,
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #3
Week #1: October 5th - October 11th. Topics: Linear independence of characters, fixed point lemma, conjugacy classes, centralizers, centers, p-groups are supersolvable, groups of order 2p are cy
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #4
Week #4: October 12th - October 18th. Topics: Composition series, Jordan-Holder's theorem, solvable groups, higher derived subgroups, nilpotent groups, the Frattini argument, rings, fields. Read
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #6
Week #6: October 26th - November 1st, and Week #7: November 9th - November 15th. Topics: Euler's totient, characteristic of a field, prime and irreducible elements, GCD's and LCM's, Euclidean PI
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #5
Week #5: October 19th - October 25th. Topics: Rings, fields, ideals, quotient rings, Chinese remainder theorem, integral domains, the field of fractions, PID's, UFD's, Euclidean domains, polynom
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #7
Week #8: November 16th - November 22nd. Topics: Algebraic extensions, transcendental elements (ex Liouville numbers), algebraically closed fields, algebraic closure, Zorn's lemma, splitting fiel
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #8
Week #9: November 23rd - November 29th. (Thanksgiving week) Topics: Perfect fields, Galois extensions, field automorphisms, the Galois group. Read: FGT [22-28] Problems due Friday, December 4th,
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #9
Week #10: November 30th - December 6th. Topics: Dedekind's inequality, finite subgroups of F are cyclic, primitive element theorem, Artin's theorem, Galois criteria, the Galois correspondence, e
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #1
Week #1: September 21st - September 27th. Topics: Fundamental theorem of algebra, roots of unity, cubic equations, Cardan's formula, unsolvability of the quintic, Newton's approximation method,
Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY
PROBLEM SET #10
Week #11: December 7th - December 13th. Topics: Normal closure, compositum, translation principle, cyclotomic fields, finite fields, normal bases, cyclic extensions, norm and trace, Hilbert's t
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 2
Rik Sengupta October 2, 2009
1. Show that Sn is generated by the tranpositions si = (i, i + 1), and that they satisfy the braid group relations. That is, si sj = sj si when |i - j|
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 3
Rik Sengupta October 9, 2009
1. Show that Q/Z is isomorphic to (C), the group of z C such that z n = 1 for some positive integer n. Is it a finitely generated abelian group? What i
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 4
Rik Sengupta October 16, 2009
1. As on page 61 in GT, just above Theorem 4.32, make a table of all possible cycle structures for S5 , and find the cardinalities of all the conjugac
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 5
Rik Sengupta October 23, 2009
1. Let F4 be a field with four elements cfw_0, 1, x, y. Write down the tables for addition and multiplication, hence showing F4 is unique. Solution. T
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 6
Rik Sengupta November 13, 2009
1. Let E be a finite field extension of F , and let R be a subring of E containing F . Show that R is necessarily a subfield of E. Solution. By the I
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 8
Rik Sengupta December 4, 2009
1. Let F be a field of characteristic p > 0, and fix an algebraic closure F . For each k > 0, show that the following subset is a subfield, Fp
-k
=
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 9
Rik Sengupta December 10, 2009
1. Let E be the splitting field of x4 - 4x2 + 2 Q[x]. Describe the automorphisms of E, and find a generator for the group Gal(E/Q). Solution. Note th
MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 10
Rik Sengupta December 18, 2009
1. Let p be a prime. Write down a normal basis for Q(p ) over Q. Now let n be any positive a integer. Is the following statement true of false: cfw_
MAT 322: Algebra with Galois Theory Claus Sorensen Problem Set 1
Rik Sengupta September 25, 2009
1. Let G be a group. Prove the generalized associative law: Given n elements a1 , . . . , an G, all ways of bracketing this ordered sequence to give it a valu