Lecture Schedule and Homework

MATH:6010:0001 - Introduction to Algebra II, Spring 2019


Book references: Dummit and Foote (D-F), Lang (L), Samuel (S) (These books have been put on reserve in the math library.)

Tuesday Thursday
Jan. 15

In Lecture:

FIELD THEORY

  • Review/notation:
    algebraic extension, irreducible polynomial, composites, translations
  • Distinguished classes of field extensions

Book reference:
  • L: V.1

HOMEWORK 1
(due at the beginning of the lecture on 01/24) 		Jan. 17

In Lecture:
  • Distinguished classes of field extensions
  • Field embeddings
  • Algebraically closed fields
  • Existence of algebraically closed fields

Book reference:
  • L: V.1, V.2
Jan. 22

In Lecture:
  • Existence of algebraic closures
  • Separable degree of a simple algebraic extension F(α)/F
  • If E/F is an algebraic extension, then every embedding of F into an algebraically closed field L can be extended to an embedding of E into L
  • Algebraic closures are unique up to isomorphism
  • Splitting fields
  • Splitting fields are unique up to isomorphism

Book reference:
  • L: V.2, V.3
Jan. 24

In Lecture:
  • Normal extensions (normal extensions are algebraic)
  • Normal extensions do not form a distinguished class
  • Normal extensions behave well under translation
  • If K/F is normal, then K/E is normal for every subextension E of K/F
  • Separable degree

Book reference:
  • L: V.3, V.4

HOMEWORK 2
(due at the beginning of the lecture on 02/07)
Jan. 29

In Lecture:
  • Multiplicativity of separable degrees
  • Separable extensions (separable extensions are algebraic)
  • Primitive element theorem

Book reference:
  • L: V.4
Jan. 31

In Lecture:
  • Primitive element theorem
  • Discussion of HOMEWORK 1
  • The class of separable extensions is distinguished
  • Separable closure
  • Normal closure

Book reference:
  • L: V.4
Feb. 05

In Lecture:
  • The separable closure of F in Fa is normal over F
  • Reminder of finite fields
  • Inseparable extensions

Book reference:
  • L: V.4, V.5, V.6
Feb. 07

In Lecture:
  • Purely inseparable extensions

Book reference:
  • L: V.6

HOMEWORK 3
(due at the beginning of the lecture on 02/21)
Feb. 12

In Lecture:

GALOIS THEORY

  • Galois extensions
  • The fundamental theorem of finite Galois theory

Book reference:
  • L: VI.1
Feb. 14

In Lecture:
  • Discussion of HOMEWORK 2
  • Galois groups of composite fields

Book reference:
  • L: VI.1
Feb. 19

In Lecture:
  • Galois groups of polynomials
  • Cyclotomic extensions
  • Linear independence of characters

Book reference:
  • L: VI.2, VI.3, VI.4
Feb. 21

In Lecture:
  • Linear independence of characters
  • The norm and the trace

Book reference:
  • L: VI.4, VI.5

HOMEWORK 4
(due at the beginning of the lecture on 03/07)
Feb. 26

In Lecture:
  • Cyclic extensions, Hilbert's Theorem 90 (multiplicative form)
  • Artin-Schreier extensions, Hilbert's Theorem 90 (additive form)
  • Galois closure of a separable extension

Book reference:
  • L: VI.6, VI.7
Feb. 28

In Lecture:
  • Discussion of HOMEWORK 3
  • Solvable and radical extensions

Book reference:
  • L: VI.7
Mar. 05

In Lecture:
  • Solvable and radical extensions

Book reference:
  • L: VI.7
Mar. 07

In Lecture:
  • A crash course on infinite limits
  • Infinite Galois extensions
Book reference:
  • L: VI.14

INTRODUCTION TO ALGEBRAIC NUMBER THEORY

  • Elements integral over a ring
Book reference:
  • S: II.1

BONUS HOMEWORK
(due at the beginning of the lecture on 04/11 -- DUE DATE HAS CHANGED!)
Mar. 12

In Lecture:
  • Elements integral over a ring
  • Integrally closed rings
  • Quadratic integer rings

Book reference:
  • S: II.1, II.2, II.5
Mar. 14

In Lecture:
  • Quadratic integer rings
  • Discussion of HOMEWORK 4

Book reference:
  • S: II.5
Mar. 19

Spring break

Mar. 21

Spring break

Mar. 26

In Lecture:
  • Norms/traces and integral elements
  • The discriminant

Book reference:
  • S: II.6, II.7
Mar. 28

In Lecture:
  • The discriminant

Book reference:
  • S: II.7, II.8

Video of lecture (Note that at the end of the lecture there is a typo, namely around 1:13:58 the norm should be the norm of K over Q and not the norm of L over K.)

IN-CLASS TEST: Thursday, March 28, 6:30-8:30PM, 106 GILH.
(see Announcements)
Apr. 02

No lecture (Prof. Bleher is at a conference)

Apr. 04

No lecture (Prof. Bleher is at a conference)

HOMEWORK 5
(due at the beginning of the lecture on 04/18)
IMPORTANT: Problem #3 was added on April 4; problems #1 and #2 are the same as previously posted on March 28.
Apr. 09

In Lecture:
  • Discussion of in-class test
  • Definition of Dedekind domains
  • Integral closure of Noetherian rings
  • Integral closure of Dedekind domains

Book reference:
  • S: III.4 (definition of Dedekind domains), III.2, III.3
Apr. 11

In Lecture:
  • Integral closure of Dedekind domains
  • Properties of prime ideals
  • Fractional ideals
  • More on Dedekind domains, unique factorization of non-zero fractional ideals

Book reference:
  • S: III.3, III.4
Apr. 16

In Lecture:
  • Dedekind domains: unique factorization of non-zero fractional ideals
  • Rings of fractions: review
  • The ring of fractions of a Dedekind domain is a Dedekind domain

Book reference:
  • S: III.4, V.1
Apr. 18

In Lecture:
  • The splitting of a prime ideal in an extension

Book reference:
  • S: V.2
Apr. 23

In Lecture:
  • Example: How p splits in Q(z) when p>2 is prime and z is a primitive p-th root of 1
  • The discriminant and ramification

Book reference:
  • S: V.2, V.3



EXTRA Lecture: Tuesday, April 23, 6:30-8:30PM, 106 GILH:
  • The discriminant and ramification
  • Discussion of HOMEWORK 5

Book reference:
  • S: V.3
Apr. 25

No lecture (Prof. Bleher is at a conference)

Apr. 30

In Lecture:
  • The splitting of a prime number in a quadratic field
  • The decomposition and inertia groups

Book reference:
  • S: V.4, VI.2
May 02

In Lecture:
  • The decomposition and inertia groups

Book reference:
  • S: VI.2

Book references: Dummit and Foote (D-F), Lang (L), Samuel (S) (These books have been put on reserve in the math library.)

Frauke Bleher
Spring 2019