Date 
Class 
Topics 
M 3/27

26
 An element over a field is algebraic if and only if it is contained in a finite extension. Degree of an element divides degree of any field extension it is contained in. An extension generated by finitely many algebraic elements is finite, of degree at most the product of the degrees of the generators. Finite simple extensions stay finite under translation.
Example: Q(2^{1/2}, 2^{1/3})/Q and Q(5^{1/2}, 2^{1/3})/Q. Realizing each as a simple extension.
Set of algebraic elements of an extension forms a field. Algebraic extensions. Example of an infinite algebraic extension: Q, the subset of elements of C that are algebraic over Q.

F 3/24

25

Degree of a field extension. Finite and infinite field extensions. Degrees multiply in towers (Theorem 6.2.5). Degree of quotient ring K[x]/⟨f(x)⟩ over a field K is the degree of f(x). Degree of an element.
Yet more group work on simple extensions.

W 3/22

24
 
M 3/20

23

Review of vector spaces: definition, subspaces, spanning sets, linearly independent sets, bases. Theorem: Every vector space that has a finite spanning set has a basis, and any two bases have the same cardinality, which we call the dimension of the vector space. (The theorem is true without the finite spanning set requirement, but the proof requires Zorn’s lemma.) If L over K is a field extension, then L is a vector space over K.
More group work on simple extensions.

F 3/17

22

Simple extensions: Suppose L = K(α) is a simple field extension of a field K. Then either α is algebraic over K, in which case the kernel of the evaluationatα homomorphism is a nonzero principal ideal generated by the minimal polynomial f(x) of α, which is irreducible, and L = K(α) = K[α] ≅ K[x]/⟨f(x)⟩. Or α is transcendental over K, in which case the evaluationatα homomorphism is injective and L = K(α) ≅ K(x), where K(x) = Q(K[x]) is the field of rational functions over K.
Group work on simple extensions.

W 3/15

21

Field extensions. Examples and nonexamples. Algebraic and transcendental elements of a field extension. Examples. Minimal polynomial of an algebraic element of a field extension. Simple extensions.

M 3/13

20

Quotient fields are unique (because they satisfy the universal mapping property for quotient fields!).
Characteristic of a ring. The characteristic of a domain is either zero or prime. Every field contains a copy either of Q or of Z_{p} for some prime p.

F 3/3

19

In a PID every nonzero prime ideal is maximal. Notation (a, b) for gcd(a, b) vs. ⟨a, b⟩ for the ideal generated by a and b. Associate elements of a ring. Quotient fields/fields of fraction: the construction (see Theorem 5.4.4) and the universal mapping property (Thereom 5.4.6).

W 3/1

18

Prime and maximal ideals. Examples in Z, field F, F[x], Z[x]. Ideal J of R is prime if and only if R/J is an integral domain, maximal if and only if R/J is a field. Maximal ideals are prime.

M 2/27

17

First isomorphism theorem for rings (or as BB calls it “fundamental homomorphism theorem for rings”): read proof of Theorem 5.2.6.
Group work: computing quotient rings:
Z/3Z,
Q[x]/⟨x^{2} − 2⟩,
Q[x]/⟨x^{2} + x + 1⟩,
Z[x]/⟨x^{2} + 1⟩,
Z[x]/⟨x⟩,
Z[x]/⟨5⟩,
Z[x]/⟨2, x⟩,
Z[x]/⟨2x⟩.
Correspondence between ideals of R/J and ideals of R containing J: read proof of Proposition 5.3.7.

F 2/24

16

Quotient rings: ring structure on R/J, universal mapping property of quotient rings.
Group work on blackboard: first isomorphism theorem for rings.

W 2/22

15

Trivial ideal and unit ideal. Intersection of ideals; sum of ideals. Ideal generated by a set of elements. A nonzero ring is a field if and only if it has exactly two ideals. Prinicipal ideal domains: Z and F[x] are PIDs, Z[x] is not a PID.

T 2/21

14

Every ring receives unique map from Z and sends unique map to the zero ring.
Review: first isomorphism theorem for abelian groups. Example: exp : R → C^{×}.
Evaluation homomorphism. Example: evaluationat√2 Q[x] → R.
Quiz feedback.

F 2/17

13

HW #2 return: problems 1, 3, 4, 6, 7, 10, 11, 14 graded.
Product of two rings: when a domain, units.
The ring isomorphism Z_{mn} ≅ Z_{m} × Z_{m} if gcd(m, n) = 1, and consequences for the Euler ϕfunction (it’s multiplicative).
The polynomial ring R[x]: domain if R is, in which case R[x]^{×} = R^{×}.
Quiz

W 2/15

12

Welldefinition of functions whose domain is naturally a set of equivalence classes (such as Z_{m} or Q). Finite integral domains are fields. Subrings. Ring homomorphisms. Images are subrings. Ideals: definitions.
HW #1 return: problems 1, 3, 4, 6, 7, 9, 10 graded.

M 2/13

11

BB 5.1: Rings, units, zero divisors, integral domains; definitions and examples.

F 2/10
W 2/8
M 2/6

10
9
8

Prof. Pollack: BB 4.3 and 4.4

F 2/3

7

Prof. Pollack: BB through end of 4.2.

W 2/1

6

Prof. Weinstein: Parts of 4.3 in BB and finite fields of size ≤ 8.

M 1/30

5

Prof. Pollack. BB through Theorem 4.2.4.

F 1/27

4

BB to end of 4.1. Also: group of units of a (commutative) ring.

W 1/25

3

BB 4.1 through Proposition 4.1.5

M 1/23

2

Symmetries of Q[√2], Q[∛2], and Q[∛2, ω].

F 1/20

1

Syllabus. Symmetries of Q[√2] preserving addition and multiplication.
