# Announcements

A sheet containing the theorems whose proofs you are responsible for on the final is posted at right. Your final is Thursday June 9, 11:30-2:30pm in our usual classroom. The final is cumulative but may focus a bit more on the topics we have covered in the second half of the quarter. Please bring a blue book.

Office hours for finals week: Corey Stone will have office hours Tuesday from 3-4pm and Wednesday from 1-2pm. Professor Rogalski will have an office hour Monday 2-3pm, and is also available by appointment.

## Basic course description

Math 100c is the third quarter of UCSD's three-quarter abstract algebra course. It is aimed to prepare students for graduate study in mathematics. The topic is Galois theory (the theory of fields). The text will be Beachy and Blair, "Abstract Algebra", 3rd edition. We will cover Chapters 6 and 8 and the end of Chapter 7.

### TA and Professor Contact Information

Professor Rogalski: office 5131 AP&M, e-mail drogalsk@ucsd.edu.

• Lecture: MWF 1pm-1:50pm, Solis 109 (note room change)
• Office hours: M 11am-12pm, W 2-3pm in 5131 AP&M

TA: Corey Stone: office 6331 AP&M, e-mail cdstone@ucsd.edu

• Section: Tu 4 pm in B412 AP&M
• Office hours: Tu 2-3pm, Th 4-5pm in 6331 AP&M

### Lecture Summaries

3/28/16: Introduction to the course. Fields. Methods larger fields for constructing fields: taking the polynomial ring over a field mod the ideal generated by an irreducible, and taking the field of fractions of an integral domain. Review of some concepts from 100b.

3/30/16: Section 6.1. Field extensions. Algebraic and transcendental elements. The minimal polynomial of an algebraic element.

4/01/16: Section 6.1-6.2. The subfield generated by a set of elements. If u is algebraic over K then K(u) is isomorphic to K[x]/(f) where f is the minimal polynomial of u over K, while if u is transcendental over K then K(u) is isomorphic to the field of rational functions in one variable over K. Vector spaces and the degree of F over K, [F:K].

4/04/16: Section 6.2. If u is algebraic over K then [K(u):K] is equal to the degree d of the minimal polynomial of u over K, and 1, u, u^2, ... u^{d-1} are a basis of K(u) as a K-vector space. If K subset E subset F, then [F:K] is infinite if and only if at least one of [E:K] or [F:E] is infinite. Otherwise, [F:K] = [E:K][F:E]. Application: if [F:K] is finite then any u in F is algebraic of degree dividing [F:K]. Examples.

4/06/16: Section 6.2. Kronecker's theorem: If f is in K[x] then there is an extension field F of K such that f has a root in F. Algebraic extensions. If K subset F is an extension then the set E of u in F which are algebraic over K is a subfield of F. The algebraic closure of Q consists of all complex numbers which are algebraic over Q. Finite degree extensions are algebraic, and finitely generated algebraic extensions have finite degree, but arbitrary algebraic extensions can have infinite degree. If K subset E subset F and E is algebraic over K and F is algebraic over E, then F is algebraic over K.

4/08/16: Section 6.3. Geometric constructions. Constructible numbers and angles. An angle theta is constructible if and only if cos theta is a constructible number. The constructible numbers form a subfield of R closed under taking square roots. lines and circles over a subfield K of R. The intersection point of two lines over K has coordinates in K, and the intersection of a circle and a line over K has coordinates in K(\sqrt{u}) for some u in K.

4/11/16: Section 6.3. An intersection of two circles over K is also the intersection of a circle over K and a line over K. The main theorem on constructibility: a real number u is construcible if and only if u is contained in Q(u_1, ..., u_n) where each u_i satisfies u_i^2 in Q(u_1, ... , u_{i-1}). Corollary: every constructible number has degree over Q which is a power of 2. Classical proofs that an arbitrary constructible angle cannot be trisected, that the cube cannot be doubled, and that the circle cannot be squared.

4/13/16: Section 6.4. What it means for a polynomial to split over an extension field. Every polynomial with complex coefficients splits over C (fundamental theorem of algebra, will prove later.) Splitting fields. Examples: splitting fields of x^3-1 and x^3 -2 over Q and their degrees over Q. Every polynomial f in K[x] has a splitting field F, and [F:K] is at most as large as (deg f)!

4/15/16: Section 6.4. Technical lemmas on splitting fields (6.4.3, 6.4.4 in the text). In particular, any two splitting fields of a polynomial f(x) over K are isomorphic via an isomorphism that restricts to the identity map on K.

4/18/16: Section 6.5. Introduction to finite fields. Examples: Z_p and its extensions. Finite fields have prime power order. Formal derivative of a polynomial. Test for a polynomial to have multiple roots. If F is a field of char p, then E = { a in F | a^{p^n} = a} is a subfield of F.

4/20/16: Section 6.5. Quick review of characteristic. The main theorem on existence and uniqueness of finite fields: for any prime power p^n a field of order p^n can be obtained by taking the splitting field of x^{p^n} - x over Z_p. Examples of the factorization of x^{p^n} - x over Z_p.

4/22/16: Section 6.5. Theorem: A finite subgroup of the multiplicative group of a field is cyclic. In particular, if F is a finite field then its multiplicative group F^{\times} is cyclic. Proof depends on two lemmas: (1) Any polynomial of degree n in F[x] has at most n roots in F; and (2) A finite abelian group G which has at most n elements of order dividing n, for all n dividing |G|, is cyclic. Application: For all n there exists an irreducible polynomial of degree n over Z_p.

4/25/16: Section 6.5-6.6. The structure of subfields of finite fields: The field GF(p^n) has a unique subfield isomorphic to GF(p^d), for every divisor d of n, and these are all of the subfields of GF(p^n). The polynomial x^{p^n} -x factors over Z_p as the product of all irreducible polynomials over Z_p whose degree divides n. The Mobius inversion formula. The formula for the number of irreducible polys of degree n over Z_p is I_p(n) = (1/n) sum_{ d | n} \mu(n/d) p^d.

4/27/16: Section 8.1. Automorphisms of a field F. The group Aut(F) of automorphisms under composition. The Galois group Gal(F/K) of automorphisms of F which fix pointwise a subfield K is a subgroup of Aut(F). Aut(Q) and Aut(R) are trivial, while Aut(C) is uncountable (only proved for Aut(Q). Gal(C/R) and Gal(Q(sqrt(2))/Q) have order 2. General lemma: Gal(F/K) permutes the set of roots in F of any polynomial with coefficients in K.

4/29/16: Review for midterm

5/2/16: Midterm

5/4/16: Section 8.1. Technical lemma: Given an isomorphism theta: K to L and splitting fields F for f(x) over K and E for theta(f(x)) over L, then the number of isomorphisms from F to E which restrict to theta on K is bounded by [F:K] and is equal to [F:K] if every irreducible factor of f(x) in K[x] has no repeated roots in F. Corollary: Gal(F/K) is bounded by [F:K] and is equal to [F:K] if that same condition holds. Examples.

5/6/16: Section 8.1-8.2. The Galois group of an extension of finite fields is cyclic. Separability. Review of the derivative test for multiple roots: f(x) has a multiple root in its splitting field if and only if gcd(f, f') > 1, where f' is the formal derivative. Form of an inseparable polynomial.

5/9/16: Section 8.2-8.3. A field of char. p is perfect if and only if every element has a pth root. Finite fields are perfect. Introduction to the fundamental theorem: the fixed field of a subgroup of the Galois group, and the Galois group of an intermediate field.

5/11/16: Section 8.3. Examples: splitting field of x^3 -2 and of x^5 -1 over Q. Correspondence between subgroups of the Galois group and intermediate fields.

5/13/16: Section 8.3. Statement of the fundamental theorem of Galois theory.

5/16/16: Section 8.3. Selected parts of the proof of the fundamental theorem of Galois theory.

5/18/16: Section 8.5. Review of roots of unity. Cyclotomic polynomials. Phi_n(x) is a monic integer polynomial of order equal to the Euler phi function of n. x^n -1 is the product of all cyclotomic polynomials Phi_d(x) as d runs over the divisors of n. Examples.

5/20/16: Section 8.5. Proof that cyclotomic polynomials are irreducible. If a regular n-gon is constructible then n is a power of two times a product of odd primes of the form 2^k + 1.

5/23/16: Section 8.4. Radical field extensions. Solvability by radicals. The Galois group of a splitting field of x^n -1 over a subfield of C is a subgroup of Z_n^{times} so is Abelian. The Galois group of the splitting field of x^n -a over a subfield of C already containing the nth roots of 1 is a subgroup of Z_n so is Abelian.

5/25/16:Section 8.4. Any radical extension can be extended to a radical extension which is a splitting field (proof omitted). Solvable groups. A polynomial over a subfield K of C is solvable by radicals if and only if its Galois group is a solvable group.

5/27/16: Section 7.6/8.4. The commutator subgroup G'. A normal subgroup H of G satisfies G/H is Abelian if and only if H contains the commutator subgroup G'. A_5 is its own commutator. A_5 is not solvable. An irreducible poly of degree 5 over Q with exactly three real roots has Galois group S_5 and is not solvable by radicals. Example of such a polynomial.

5/30/16: Memorial Day (No class)

6/01/16: Section 8.2: Algebraically closed fields. Different characterizations of algebraically closed fields. The fundamental theorem of algebra: the field C of complex numbers is algebraically closed. Prove of the fundamental theorem using Galois theory and a few facts from analysis.

6/03/16: Review for final