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.

Last revised on July 6, 2015 at 13:44:56. See the history of this page for a list of all contributions to it.