Study guide for final exam
You should know:
Everything from the first study guide.
- The division algorithm and its consequences, in particular that
F[x] is a PID if F is a field.
- Irreducible/reducible polynomials. Tests for irreducibility:
- Degree 2 and 3 test
- mod p reduction test
- Eisenstein's criterion
- Irreducible/prime elements of an integral domain. Relation
with prime ideals.
- Euclidean domain ==> PID ==> UFD. Example of a UFD
which is not a PID.
- Vector spaces. Linearly independent and spanning
sets. Bases; invariance of
size of a basis.
- Field extensions. An element of an extension K of F is
algebraic/transcendental
over F if...
- Degree of an extension; degree is multiplicative.
- p(x) the minimal polynomial of a over F implies F(a) isomorphic
to F[x]/<p(a)>.
In that case, [F(a): F] = deg p(x). The minimal poly is
irreducible; any other polynomial
with a as a root is a multiple of p(x).
- Splitting fields; splitting fields exist and are unique.
- Finite fields: examples. The main classification theorem
22.1. The group of units is
cyclic. You should be able to calculate in small finite
fields.
Note that we skipped pp.354-357, as well as from Theorem 21.6
(Primitive
element theorem)
to the end of chapter 21, and the last part of chapter 22 ("Subfields
of a finite field"). Material
from these parts will not appear on the exam.
Some study questions:
p. 332 -- 21, 25, 27, 30, 31
Chapter 20 -- 14, 16, 27, 34
Chapter 21 -- 8, 10, 14, 15, 16, 26, 33
Chapter 22 -- 5, 14, 27