21238 Lecture Notes - Lecture 1: Topological Vector Space, Free Module, Module Homomorphism

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Published on 13 Aug 2017
School
CMU
Department
Mathematical Sciences
Course
21238
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 ModuleHomomorphisms............................................ 5
3 January 23, 2017 6
3.1 GenerationofModules............................................. 6
3.2 FreeModules .................................................. 6
4 January 25, 2017 7
4.1 FreeModulescont................................................ 7
5 January 27, 2017 8
5.1 FreeModulescont................................................ 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 FiniteDimensionalVectorSpaces....................................... 12
10 February 8, 2017 13
11 February 10, 2017 14
12 Feburary 13, 2017 15
12.1TensorProducts ................................................ 15
13 February 15, 2017 16
14 February 17, 2017 16
14.1ExactSequences ................................................ 16
14.2Splitting..................................................... 17
15 February 20, 2017 17
15.1TopologicalVectorSpaces ........................................... 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
18.1RepresentationTheory............................................. 19
18.2 Decomposition of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
19 March 1, 2017 21
19.1MaschkesTheorem............................................... 21
20 March 3, 2017 22
20.1Artin-WedderburnTheorem.......................................... 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
24.1ExtensionsofFields .............................................. 25
25 March 24, 2017 26
25.1ExtensionsofFields(cont.) .......................................... 26
25.2 Straight Edge and Compass Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
26 March 27, 2017 27
26.1AlgebraicExtensions.............................................. 27
26.2SplittingFields ................................................. 28
27 March 29, 2017 28
27.1AlgebraicClosures ............................................... 28
28 March 31, 2017 29
28.1LinearOrders.................................................. 29
28.2AlgebraicClosures(cont.)........................................... 29
29 April 3, 2017 30
29.1SeparablePolynomials............................................. 30
30 April 5, 2017 31
31 April 7, 2017 31
31.1GaloisExtensions................................................ 31
32 April 10, 2017 33
32.1GaloisExtensions(cont.) ........................................... 33
33 April 12, 2017 34
33.1FunwithGaloisTheory ............................................ 34
33.1.1 NormalSubgroups ........................................... 34
33.1.2 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
34 April 14, 2017 35
35 Radical Extensions 35
35.1SolvableGroups ................................................ 35
36 April 17, 2017 36
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21-238 Math Studies Algebra II
37 April 19, 2017 36
37.1TheConverseofGalois............................................. 36
38 April 24, 2017 37
39 April 26, 2017 38
39.1Ultralters(cont.) ............................................... 38
39.2SomeCombinatorics .............................................. 39
40 April 28, 2017 39
40.1TheSpaceofUltralters............................................ 39
40.1.1 CrashCourseinTopology....................................... 39
40.1.2 Topology on Space of Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
41 May 1, 2017 40
41.1Semigroups ................................................... 40
42 May 3, 2017 41
42.1MinimalSubsemigroups ............................................ 41
43 May 5, 2017 42
43.1HindmansTheorem .............................................. 42
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Document Summary

6. 1 finitely generated modules over principle ideal domains . 1. 1 logistics: o ce: wean 7121, o ce hours: tuesday 1:30 - 3:00, andrew id: clintonc, grading: 20% homework (graded for completeness), 20% 2 midterms, 40% nal. A module is something for a ring to act on. Informally, modules are the equivalent of group actions for rings. In this course, a ring is not commutative and they don"t necessarily have an identity. Let r = (r, +r, r) be a ring with identity. If m is a unital z-module, we have that. 1 m = m = 2 m = (1 + 1) m = m + m. More generally, for z z, z times z m = m + + m. Thus, the z-action is determined completely by the group (m, +m ). In summary, abelian groups and z-modules are the same thing. Let r = r and consider the euclidean plane r2 = {(x, y) : x, y r}.