Polynomial Rings Flashcards
Remainder theorem.
Let F be a field, a in F, and f(x) in F[x]. Then f(a) is the remainder of f(x) divided by x-a.
Principal ideal domain?
An integral domain R in which every ideal has the form
<a> = {ra | r in R} for some a in R. </a>
If F is a field, what is F[x]?
Principal ideal domain and thus UFD.
Content of a polynomial?
The gcd of the coefficients.
Primitive polynomial?
Element of Z[x] with content 1.
Mod p irreducibility test.
Let p be prime and f(x) in Z[x] with
deg >= 1. Let f’(x) be the polynomial in Z_p[x] obtained by reducing the coefficients of f(x). If f’(x) is irreducible over Z_p[x] and deg(f’(x))=deg(f(x)) then f(x) is irreducible over Q.
Eisenstein’s criterion.
For f(x) in Z[x], if there is a prime p such that p does not divide a_n, but p|a_i for all i
F[x]/<p></p>
Is a field if p(x) is irreducible over F.
Associates?
Elements a and b of integral domain D are associates of a=ub where u is a unit.
Irreducible?
An element a of an integral domain D is irreducible if a is nonzero, not a unit, and if for b,c in D, if a=bc, then either b or c is a unit.
Prime?
A nonzero nonunit element of an integral domain D such that a|bc implies a|b or a|c.
In an integral domain, every prime is…?
When are they equivalent?
Irreducible.
In a principal ideal domain.
Unique factorization domain?
An integral domain D such that every nonzero non-unit element can be written as a product of irreducibles. Furthermore, the factorization is unique up to associates.
Every principal ideal domain is…?
UFD
Euclidean domain?
An integral domain D if there is a measure function, d, such that
d(a)
