What does the universal property of $K[X]$ say?

For a $K$-algebra and chosen element $a$, there is a unique homomorphism from $K[X]$ fixing $K$ and sending $X$ to $a$.

Can distinct polynomials define the same function over a finite field?

Yes. Over a field with $q$ elements, $0$ and $X^q-X$ define the same zero function but are distinct polynomials.

What is the minimal polynomial of an algebraic element over a field?

The unique monic polynomial of least degree that vanishes at the element; it is irreducible.

When is $K[X]/(p)$ a field, for $p\in K[X]$ over a field $K$?

Exactly when $p$ is irreducible.

How is the total degree of a multivariate monomial determined?

Add the exponents in its exponent vector; for example, $X^2Y^3$ has total degree $5$.

Abstract algebraLesson 2 of 2 in the course Rings and ideals

Polynomial Rings: Evaluation, Factorization, Quotients, and Multivariate Structure

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

This lesson introduces the polynomial ring $K[X]$ : what its elements are, how their coefficients and indeterminate work, and how addition and multiplication are performed.

Let $K$ be a field or, more generally, a commutative ring. The univariate polynomial ring $K[X]$ consists of finite expressions such as

where each coefficient $p_i$ belongs to $K$ . The symbol $X$ is an indeterminate. Its powers follow the usual exponent rules, including $X^0=1$ and $X^kX^l=X^{k+l}$ .

All flashcards

What does the universal property of $K[X]$ say?: For a $K$ -algebra and chosen element $a$ , there is a unique homomorphism from $K[X]$ fixing $K$ and sending $X$ to $a$ .
Can distinct polynomials define the same function over a finite field?: Yes. Over a field with $q$ elements, $0$ and $X^q-X$ define the same zero function but are distinct polynomials.
What is the minimal polynomial of an algebraic element over a field?: The unique monic polynomial of least degree that vanishes at the element; it is irreducible.
When is $K[X]/(p)$ a field, for $p\in K[X]$ over a field $K$ ?: Exactly when $p$ is irreducible.
How is the total degree of a multivariate monomial determined?: Add the exponents in its exponent vector; for example, $X^2Y^3$ has total degree $5$ .

Quiz questions

For polynomials $a,b\in K[X]$ over a field, with $b\ne 0$ , what does the division algorithm guarantee?Answer: A quotient and remainder, with the remainder zero or of degree less than $b$ .
For a polynomial $p\in K[X]$ over a field, when is the quotient ring $K[X]/(p)$ a field?Answer: When $p$ is irreducible over $K$ .
What does the universal property of $K[X]$ guarantee for a chosen element $a$ of a $K$ -algebra?Answer: A unique homomorphism fixing $K$ and sending $X$ to $a$ .
In a field with $q$ elements, what is true of the polynomial $X^q-X$ when evaluated at field elements?Answer: It evaluates to zero at every element of the field.
How unique is irreducible factorization for a nonconstant polynomial over a field?Answer: Unique up to the order of factors and multiplication by units.

Based on the Wikipedia article Polynomial ring, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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