## Introduction to Modern Algebra II

#### Mathematics GU4042 Prof. Michael ThaddeusSpring 2017

REVIEW SESSION: Saturday, May 6 at 4:10 pm in 602 Hamilton. Bring your questions, which are the only agenda.

FINAL EXAM: Tuesday, May 9 from 4:10-7 pm in 602 Hamilton.

Assignments are due in the collection box marked "Modern Algebra II" outside 417 Mathematics. Make sure you use the right box!

Assignment #1, due Friday, January 27 at 5 pm.
Assignment #2, due Friday, February 3 at 5 pm.
Assignment #3, due Friday, February 10 at 5 pm.
Assignment #4, due Friday, February 17 at 5 pm.
Assignment #5, due Friday, February 24 at 5 pm.
Assignment #6, due Friday, March 10 at 5 pm.
Assignment #7, due Friday, March 24 at 5 pm.
Assignment #8, due Friday, March 31 at 5 pm.
Assignment #9, due Friday, April 7 at 5 pm.
Assignment #10, due Friday, April 21 at 5 pm.
Assignment #11, due Monday, May 1 at 5 pm.
Alternative hints for A11#5b: You may or may not wish to use 5a.
Let E be the splitting field of xp - a over F. Show that the splitting field K of xp -1 is a subfield of E.
Then show that (a) if K = E, then xp - a has a root in F; (b) if not, then xp - a is irreducible over K, hence over F.

Tentative list of topics
A table is given below of topics to be covered in the course, together with relevant sections of both recommended texts. Several disclaimers apply.
(1) The presentation in lecture will not closely follow either text and may include facts presented in other sections, or not at all, in one text or the other.
(2) The last few topics may or may not be covered, depending on timing.
(3) The lecturer reserves the right to permute or alter the topics at any time!

Topic Fraleigh Dummit & Foote
RINGS
Monoids, rings, and fields 5.1 7.1, 7.2
Examples 5.1 7.2
Ring homomorphisms 6.1 7.3
Subrings and ideals 6.1 7.4
Integral domains 5.2 7.1
Maximal and prime ideals 6.2 7.4
Irreducible and prime elements 8.3
The division algorithm 5.6 9.2
Euclidean and principal ideal domains 7.2 8.2, 8.4
F a field implies F[x] a PID 6.2 9.2
Gaussian integers 7.3 8.1
Euclidean algorithm and GCD 7.2 8.1
Irreducible and prime elements in PIDs 7.1 8.3
Unique factorization domains 7.1 8.3
PID implies UFD 7.1 8.3
Existence of GCDs in UFDs 7.1 8.3
Gauss's lemma 7.1 9.3
R a UFD implies R[x] a UFD 7.1 9.3
Eisenstein's criterion 5.6 9.4
Prime numbers as sums of squares 7.3 8.3
FIELDS
Algebraic elements; minimal polynomial 8.1 13.1, 13.2
Multiplicativity of degrees of field extensions 8.3 13.2
Ruler and compass constructions 8.4 13.3
Existence of splitting fields 9.3 13.4
Repeated roots and the derivative 9.4 13.5
Finite fields 8.5 13.5
Separability, perfect fields 9.4, 9.5 13.5
Automorphisms of splitting fields 9.1 14.1
The extension theorem 9.2 13.4
The Galois group 9.6 14.1
Automorphisms of fields and their fixed points 9.1, 9.6 14.1
Artin's lemma: [E:F] ≤ # G
Normal and Galois extensions 9.6 13.4, 14.1
Equivalent conditions to be Galois 9.6 14.2
The fundamental theorem of Galois theory 9.6 14.2
Cyclotomic extensions 9.7, 9.8 13.6, 14.5
Primitive elements 9.4 14.4
Quadratic and other cyclic extensions 14.5
Kummer's theorem on cyclic extensions 14.7
Radical and solvable extensions 9.9 14.7
Insolvability of the general quintic 9.9 14.7