MA 542 A1 — Fall 2019
Abstract Algebra
REVIEW SHEET
Definitions
- ring, ring with unity, commutative ring
- ring homomorphism
- ring isomorphism
- divisors of 0
- integral domain
- characteristic of a ring
- equivalence relation
- irreducible polynomials over F
- ideal
- R/I where I is an ideal (defn of addition and multiplication)
- prime ideal
- maximal ideal
- principal ideal
Theorems from class that you should know how to prove:
- 0 * a = 0
- a(-b) = -(ab)
- in an integral domain cancellation holds.
- fields are integral domains.
- finite integral domains are fields.
- Fermat's little theorem.
- addition and multiplication are well-defined in the field of quotients.
- f(a) = 0 implies f(x) = (x-a)g(x)
- degree f(x) = 2,3. Then f(x) is irreducible iff f(x) has no roots.
- Eisenstein criteria
- multiplication is well-defined in R/I where I is an ideal.
- all ideals in Z are principal.
- all ideals in F[x] are principal (F = field).
- R/I is an integral domain iff I is a prime ideal.
- R/I is a field iff I is a maximal ideal.
- p|ab implies p|a or p|b for p prime
- f(x)|g(x)h(x) implies f(x)|g(x) or f(x)|h(x) for f(x) irreducible
- pZ is maximal iff p is prime
- f(x)F[x] is maximal iff f(x) is irreducible