# Spring 2023

### Time and Place

• Lectures: MWF 12:05pm – 12:55pm Carver 132

### Examination

Each week a homework set is due, except first week and the week of an exam. There will be two non-cumulative take-home exams and no final.

The first exam will be made available on March 3, 2023 and is due a week later in class. It is open book and open notes, but not open internet. And it should be solved individually.

### Homework Sets

1. (due Fri 1/27) 1. Let $$f:R\to S$$ be a ring homomorphism. (a) Prove that if $$M$$ is an $$S$$–module then $$M$$ becomes an $$R$$–module by defining $$r.m=f(r).m$$. (b) Prove that if $$\varphi:M\to N$$ is an $$S$$‐module homomorphism, then $$\varphi$$ is automatically an $$R$$–module homomorphsm, when $$M$$ and $$N$$ are turned into $$R$$–modules as in part (a). §10.1: 9, 19, 21.
2. (due Fri 2/3) §10.2: 9, 13 (Hard. First assume $$I^2=0$$); §10.3: 7, 9; §10.4: 6.
3. (due Fri 2/10) §10.4: 11, 16, 22, 24, 25.
4. (due Fri 2/17) §10.5: 10(a), 11(a), 12(a), Appendix II.1: 4 (Typo: $$\to$$ should say $$\mapsto$$).
5. (due Fri 2/24) 1. Let $$X,A$$ be abelian groups. Prove TFAE: (i) $$\mathrm{Hom}_\mathbb{Z}(X,A)=0$$ (ii) for every abelian group $$B$$ and $$\mathbb{Z}$$–bilinear map $$\beta:X\times B\to A$$ we have $$\beta(x,b)=0$$ for all $$x\in X$$ and $$b\in B$$ [Hint: Use $$\otimes$$–Hom adjunction]; §10.5: 1(b)(e), 2(a), 14(a), 27(a).
6. (due Fri 3/3) 1. Show that a module $$P$$ is projective if and only if there are elements $$x_i\in P$$ and $$\xi_i\in\mathrm{Hom}_R(P,R)$$ such that $$a=\sum_i \xi_i(a)x_i$$ for all $$a\in P$$; §10.5: 3, 6.
7. (due Fri 3/31) §11.5: 5 (assume $$R$$ is a field), 13; §13.1: 1 (recall Eisenstein's Criterion), 4.
8. (due Fri 4/7) §13.2: 7, 10, 14, 16.
9. (due Fri 4/14) 1. Let $$\epsilon=\cos(\pi/9)$$. Show that $$[\mathbb{Q}(\epsilon):\mathbb{Q}]=3$$ (ask for hint if needed). (As discussed in class, this shows that the angle $$\pi/3$$ cannot be trisected.); 2. Lindemann proved in 1882 that $$\pi$$ is transcendental over $$\mathbb{Q}$$. Assuming this result, show that "squaring the circle" is impossible: There is no construction by straightedge and compass that will produce from a radius of a given circle, the side of a square that has the same area as the circle; §13.3: 4, 5.
10. (due Fri 4/21) §13.4: 1, 3, 5.
11. (due Fri 4/28) §13.5: 1, 2; §13.6: 6; §14.1: 4, 5. Read about the life of Galois.
12. (due Fri 5/5) §14.2: 4, 10, 14. (Note: Problem 16 has been moved to take-home exam 2.)

### Lecture Summary

#### Part I: Modules

Chapters 9–12

In the first part of the course we study modules. These are generalizations of abelian groups and vector spaces and play an important role in representation theory, differential geometry, and other areas. Beyond the basics, key goals include:

• tensor products and multilinear algebra
• finitely generated modules over PIDs (with applications to normal forms of matrices)
• some category theory to help us see the unity between constructions

Below is a short summary of each lecture.

1. Definition of modules. Examples.
2. More examples. Algebras. Group representations. Submodules. Direct sums (coproducts) and products.
3. Universal property of products and coproducts. Free modules.
4. Quotient modules and isomorphism theorems.
5. Presentations of modules. Restriction and induction (examples/motivation).
6. Free modules, definition of induced modules $$S \otimes_R M$$.
7. Tensor products of general bimodules. Balanced maps and the universal property of tensor products.
8. Examples: Computing $$\mathbb{Z}_n\otimes_\mathbb{Z}\mathbb{Z}_m$$. The "tensor unit" isomorphism $$R\otimes_R M\cong M$$.
9. Distributive law for tensor products and coproduct. Corollary for free modules. Representation of $$S_3$$ by induction from the sign-representation of $$S_2$$.
10. Definition of Category. Examples.
11. More examples of categories. Functors.
13. Exact sequences.
14. The short five lemma. Split short exact sequences.
15. Exact functors. The Hom functor is left exact. Projective modules.
16. Characterization of projective modules. Examples.
17. Injective modules.
18. Flat modules.
19. Noetherian modules and rings. PIDs are Noetherian.
20. Fundamental Theorem of Finitely Generated Modules over PIDs
21. Application to Jordan-Chevalley Canonical Form.
22. The Tensor Algebra.
23. Symmetric and Exterior Algebras. Determinant.

#### Part II: Fields

Chapters 13–14

In the second part of the course we study fields. Since every homomorphism between fields is injective, one is led to study subfields and field extensions. Galois theory provides a link from this to group theory. Historically, field theory led to the resolution of several problems in Euclidean geometry that had been open for thousands of years.

Below is a short summary of each lecture.

1. Fields. Constructing new fields from old ones. Prime subfield. Idea of invariants.
2. Characteristic of a field. Field extensions. Adjoining a root. Degree.
3. Simple and algebraic field extensions.
4. Algebraic vs. Finite vs. Finitely Generated field extensions. Set of algebraic elements is closed under field operations.
5. The algebraic closure of a subfield in a field. Composite extensions. Constructions by straightedge and compass.
6. More on constructions by straightedge and compass.
7. The field of constructible numbers. Impossibility of certain geometric constructions.
8. Splitting fields; existence.
9. Uniqueness of splitting fields.
10. Algebraically closed fields. Existence and uniqueness of the algebraic closure of a field.
11. Cyclotomic polynomials and cyclotomic extensions.
12. Irreducibility of the cyclotomic polynomial. Automorphism group of a field extension.
13. Fixed fields. Examples of Galois correspondence.
14. Order of automorphism group for splitting fields.
15. First half of Galois correspondence: $$\Gamma\circ\Phi=\mathrm{Id}$$ for finite extensions $$K/F$$.
16. (planned) Second half of Galois correspondence: $$\Phi\circ\Gamma=\mathrm{Id}$$ for finite normal separable field extensions $$K/F$$.
17. (planned) Examples, cyclic extensions, symmetric group.
18. (planned) Radicals and solvable Galois groups; insolvability of the quintic.

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