-
(Fraleigh 23.1) Division Algorithm Let
with for and and . Then there are unique polynomials and such that Where either
or -
Intuition: Either the remainder
or it is not. If it is not, and assuming , it is possible to find a smaller remainder ad infinitum, leading to the contradiction. -
(Fraleigh e23.36) Remainder Theorem Let
where is a field. Let . Then divided by has remainder . -
(Fraleigh 28.6) Let
such that . The common zeros in of and are the same as the common zeros of and . Also, the common divisors in
of and are the same as the same as the common divisors of and .
-
-
(Fraleigh 23.3) Factor Theorem. An element
is a zero of if and only if is a factor of in . -
(Fraleigh 23.5) A nonzero polynomial
of degree can have at most zeros in field . -
A nonconstant polynomial
is irreducible over (a.k.a. an irreducible polynomial over ) if cannot be expressed as a product of two polynomials both in lower degree than . Otherwise,
is reducible - A polynomial can be irreducible in field
but reducible in a larger field .
- A polynomial can be irreducible in field
-
(Fraleigh 23.10)
is reducible over if and only if it has a zero in . -
(Frraleigh 23.18, Fraleigh 27.27) Let
be an irreducible polynomial in . If divides for then either divides or divides -
(Fraleigh 23.19) If
is irreducible in and divides the product for then divides for at least one -
(Fraleigh 23.20) If
is a field, then every nonconstant polynomial can be factored in into a product of irreducible polynomials. The irreducible polynomials are unique up to order and non-zero constant factors in .
Special Cases
-
(Fraleigh 23.11) Gauss Lemma If
then factors into a product of two polynomials of lower degrees if and only if it has such a factorization with polynomials of the same degrees and in . -
(Fraleigh 23.12) If
with and if has a zero in , then it has a zero and must divide . Another way to say this: The zeros of a polynomial (in the field of integers) are rational and must divide the constant term.
-
(Fraleigh 23.15) Eisenstein Criterion Let
be prime. Suppose that . Then is irreducible over if the following hold: . Then is irreducible over .
-
The set of cyclotonic polynomials are irreducible over
for any prime. They are defined as