irreducible polynomial

A non-zero polynomial ff with coefficients in a field kk is irreducible if when written as the product ghg h of two polynomials, one of gg or hh is a constant (and necessarily non-zero) polynomial. Equivalently, a polynomial ff is irreducible if the ideal it generates is a maximal ideal of the polynomial ring k[x]k[x].

In other words, a polynomial ff is irreducible if it is an irreducible element of k[x]k[x] as an integral domain.

Notice that under this definition, the zero polynomial is not considered to be irreducible. An alternative definition, which applies to the case of coefficients in a commutative ring RR, is that a polynomial ff is irreducible if, whenever ff divides ghg h, either ff divides gg or ff divides hh. Under this definition, a polynomial is irreducible if it generates a prime ideal in R[x]R[x], and the zero polynomial is irreducible if RR is an integral domain.

If LL is a field extension of kk and αL\alpha \in L is algebraic over kk (i.e., the smallest subextension containing α\alpha, k(α)k(\alpha), is finite-dimensional over kk), then the kk-algebra map k[x]k(α):xαk[x] \to k(\alpha): x \mapsto \alpha has a non-trivial kernel which is a maximal ideal, which being principal is generated by an irreducible polynomial ff. The unique monic polynomial generator is called the irreducible polynomial of α\alpha.

Revised on July 6, 2015 13:44:56 by Todd Trimble (