Polynomial Rings

I took Abstract Algebra I/II in UW-Madison(MATH541) with professor Chenxi Wu and Tonghai Yang. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

RECAP

Please read this introduction with a complete understanding of ring and concepts including Euclidean / Principal Ideal / Unique Factorization Domains.

Definitions and Basic Concepts

For a given indeterminate with coefficients from a given ring , the polynomial ring is the polynomials of the form , with and . For this polynomial, is its degree, is its leading term, is its leading coefficient. The polynomial is monic if .

The addition in is

and the multiplication in is

In this way is a well-defined communicative ring with identity . Clearly is a subring of .

Proposition. Let be an integral domain, then

1. degree = degree + degree if and are both nonzero;
2. the units of are units of ;
3. is an integral domain.

Proposition. Let be an ideal of the ring and let . Then

Definition. [Polynomial ring in the variables] is defined inductively by

Polynomial Rings Over Fields

Theorem. Let be a field, the polynomial ring is a Euclidean Domain.

Colloary. If is a field, then is a PID and UDF.

Polynomial Rings that are UFD

Proposition. (Gauss' Lemma) Let be a UFD with field of fractions , then if then is reducible in . Precisely, if for some nonconstant polynomials , then there are nonzero elements such that and both lie in and is a factorization in .

Corollary. Let be a UFD. Let be its field of fractions and let . Suppose the gcd of all its coefficients are , then is irreducible in if and only if it's irreducible in .

Theorem. is a UDF if and only if is a UFD.

Corollary. If is a UFD, then a polynomial ring in an arbitrary number of variables with coefficients in is also a UFD.

Irreducibility Criteria

Proposition. Let be a field and let . Then has a factor of degree one if and only if has a root in . That is, there is an with .

Proposition. A polynomial of degree two or three over a field is reducible if and only if it has a root in .

Proposition. Let be a polynomial of degree with integer coefficients. If and are relatively prime and is a root of (that is, ), we have and .

Proposition. Let be a proper ideal in the integral domain . If the image of in cannot be factored in into two polynomials of smaller degree, then is irreducible in .

Proposition. [Eisenstein's Criterion] Let be a prime ideal of the integral domain and let be a polynomial in . If are all elements of , and , then is irreducible in .