Abstract algebra MOC

Polynomial ring

Let be a ring. A polynomial in the indeterminate and with coëfficients in is a finite linear combination of nonnegative powers of with coëfficients in :¹² ring

where has finite support³, hence it may be viewed as an element of the free module . This free module forms the polynomial ring with the structure of a ring (and K-monoid) given by the Monoid ring construction, thus

The leading term of a polynomial is the term with the largest exponent , and the coëfficient is called the degree . We write .

• A polynomial with leading coëfficient one is called a monic polynomial (not to be confused with monic).
• A polynomial is irreducible if has no divisors other than itself and (similar to prime numbers), however a polynomial can often be reduced by looking at a bigger underlying ring, for examples can only be factorised using the complex numbers.

A polynomial in multiple indeterminates may be formed by iterating the above process, so .

Universal property

An fundamental property of a polynomial ring is that the elements are in the centre. The polynomial ring is characterized up to unique isomorphism by the following universal property:

Let be a ring. The polynomial ring is a pair consisting of a ring and a ring homomorphism such that is an element of the centre and given any ring homomorphism and element of the centralizer , then there exists a unique ring homomorphism such that the following diagram commutes

and .

Evaluation map

Let and . By the above construction, there exists a unique ring homomorphism such that and , which is called the evaluation map at .

Properties

• The polynomial ring over an integral domain is an integral domain
• The polynomial ring over a field is a Euclidean domain
• Polynomial ring over a noetherian ring is noetherian (Hilbert’s basis theorem)
• Polynomial ring over a UFD is a UFD

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Footnotes

1. 2009. Algebra: Chapter 0, §III.1.3, pp. 124ff. ↩

2. 2017. Contemporary abstract algebra, §16, pp. 276ff. ↩

3. cf. Series ring ↩