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Study Guide

21238- Midterm Exam Guide - Comprehensive Notes for the exam ( 48 pages long!)


Department
Mathematical Sciences
Course Code
21238
Professor
Clinton Conley
Study Guide
Midterm

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CMU
21238
MIDTERM EXAM
STUDY GUIDE

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Math Studies Algebra II
Prof. Clinton Conley
Spring 2017
Contents
1 January 18, 2017 4
1.1 Logistics..................................................... 4
1.2 Modules..................................................... 4
2 January 20, 2017 5
2.1 Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Module Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 January 23, 2017 6
3.1 Generation of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 January 25, 2017 7
4.1 Free Modules cont. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 January 27, 2017 8
5.1 Free Modules cont. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2 Dimension.................................................... 9
6 January 30, 2017 9
6.1 Finitely Generated Modules over Principle Ideal Domains . . . . . . . . . . . . . . . . . . . . . . . . . 9
7 February 1, 2017 11
8 February 3, 2017 12
9 February 6, 2017 12
9.1 Finite Dimensional Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
10 February 8, 2017 13
11 February 10, 2017 14
12 Feburary 13, 2017 15
12.1 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
13 February 15, 2017 16
14 February 17, 2017 16
14.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
14.2Splitting..................................................... 17
15 February 20, 2017 17
15.1 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
16 February 22, 2017 18
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21-238 Math Studies Algebra II
17 February 24, 2017 18
18 February 27, 2017 19
18.1 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
18.2 Decomposition of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
19 March 1, 2017 21
19.1 Maschke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
20 March 3, 2017 22
20.1 Artin-Wedderburn Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
21 March 6, 2017 23
22 March 8, 2017 24
22.1 Examples Using C[G] ............................................. 24
23 March 20, 2017 25
24 March 22, 2017 25
24.1 Extensions of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
25 March 24, 2017 26
25.1 Extensions of Fields (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
25.2 Straight Edge and Compass Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
26 March 27, 2017 27
26.1 Algebraic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
26.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
27 March 29, 2017 28
27.1 Algebraic Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
28 March 31, 2017 29
28.1 Linear Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
28.2 Algebraic Closures (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
29 April 3, 2017 30
29.1 Separable Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
30 April 5, 2017 31
31 April 7, 2017 31
31.1 Galois Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
32 April 10, 2017 33
32.1 Galois Extensions (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
33 April 12, 2017 34
33.1 Fun with Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
33.1.1 Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
33.1.2 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
34 April 14, 2017 35
35 Radical Extensions 35
35.1 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
36 April 17, 2017 36
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