Polynomial ring

Let $𝑅$ be a ring. A polynomial $𝑓 (𝑥)$ in the indeterminate $𝑥$ and with coëfficients in $𝑅$ is a finite linear combination of nonnegative powers of $𝑥$ with coëfficients in $𝑅$ :¹² #m/def/ring

𝑓 (𝑥) = \infty \sum 𝑖 = 0 𝑓 𝑖 𝑥 𝑖

where $𝑓 (-)$ has finite support³, hence it may be viewed as an element of the free module $𝑅 [ℕ 0]$ . This free module forms the polynomial ring $𝑅 [𝑥]$ with the structure of a ring (and $𝕂$ -monoid) given by the Monoid ring construction, thus

𝑥 𝑛 \cdot 𝑥 𝑚 = 𝑥 𝑛 + 𝑚

The leading term of a polynomial is the term $𝑓 𝑛 𝑥 𝑛$ with the largest exponent $𝑛$ , and the coëfficient $𝑓 𝑛$ is called the degree $d e g 𝑓$ . We write $d e g 0 = - \infty$ .

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 $1$ (similar to prime numbers), however a polynomial can often be reduced by looking at a bigger underlying ring, for examples $𝑥 2 + 1$ can only be factorised using the complex numbers.

A polynomial in multiple indeterminates may be formed by iterating the above process, so $𝑅 [𝑥, 𝑦, 𝑧] = 𝑅 [𝑥] [𝑦] [𝑧]$ .

Universal property

An fundamental property of a polynomial ring is that the elements ${𝑥 𝑛} \infty 𝑛 = 1$ are in the centre. The polynomial ring $𝑅 [𝑥]$ is characterized up to unique isomorphism by the following universal property:

Let $𝑅$ be a ring. The polynomial ring is a pair consisting of a ring $𝑅 [𝑥]$ and a ring homomorphism $𝜄 𝑅 : 𝑅 \to 𝑅 [𝑥]$ such that $𝑥$ is an element of the centre $𝑍 (𝑅 [𝑥])$ and given any ring homomorphism $𝑓 : 𝑅 \to 𝑄$ and element $𝑞$ of the centralizer $𝐶 (𝑓 (𝑅))$ , then there exists a unique ring homomorphism $¯ 𝑓 : 𝑅 [𝑥] \to 𝑄$ such that the following diagram commutes

and $¯ 𝑓 (𝑥) = 𝑞$ .

Evaluation map

Let $𝑓 (𝑥) \in 𝑅 [𝑥]$ and $𝑟 \in 𝑅$ . By the above construction, there exists a unique ring homomorphism $𝜖 (𝑟) : 𝑅 [𝑥] \to 𝑅$ such that $𝜖 (𝑟) 𝜄 𝑅 = i d 𝑅$ and $𝜂 (𝑟) (𝑥) = 𝑟$ , which is called the evaluation map at $𝑟$ .

Properties

#state/tidy | #lang/en | #SemBr

2009. Algebra: Chapter 0, §III.1.3, pp. 124ff. ↩
2017. Contemporary abstract algebra, §16, pp. 276ff. ↩
cf. Series ring ↩

Polynomial ring

Universal property

Evaluation map

Properties

Footnotes