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Section Daily Prep 18

We will discuss divisibility in polynomial rings, including tests for irreducibility of polynomials. We will also see that the division algorithm for polynomials implies that a polynomial ring over a field is a PID.

Objectives: Basic Learning Objectives

Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.

State and instantiate the definition of: irreducible polynomial, reducible polynomial, content, primitive polynomial
State and apply the following mathematical results: Reducibility Test for Degree 2 and 3 Polynomials

Subsection Resources for Learning

Use these resources to prepare for class and answer the questions below.

Gallian, Chapter 17, pp. 311-312

Figure 179. Reference Video for Irreducible Polynomials

Subsection Important Terms

Definition 180. Irreducible and Reducible Polynomials.

Let \(D\) be an integral domain. A polynomial \(f(x)\in D[x]\) that is neither the zero polynomial nor a unit is said to be irreducible over \(D\) if, whenever \(f(x)\) is expressed as a product \(f(x)=g(x)h(x)\text{,}\) with \(g(x)\) and \(h(x)\) from \(D[x]\text{,}\) then \(g(x)\) or \(h(x)\) is a unit in \(D[x]\text{.}\) A nonzero, nonunit polynomial in \(D[x]\) that is not irreducible over \(D\) is called reducible over \(D\).

Theorem 181. Reducibility Test for Degree 2 or 3 Polynomials.

Let \(F\) be a field. If \(f(x)\in F[x]\) has degree \(2\) or \(3\text{,}\) then \(f(x)\) is reducible over \(F\) if and only if \(f(x)\) has a zero in \(F\text{.}\)

Definition 182. Primitive Polynomials.

Let \(f\in \Z[x]\text{.}\) The content of \(f\text{,}\) denoted \(c(f)\text{,}\) is the greatest common divisor of the coefficients of \(f\text{.}\) A polynomial \(f\in \Z[x]\) is said to be primitive if \(c(f)=1\text{.}\)