Irreducible polynomial

A polynomial $f=f(x_1,\ldots,x_n)$ in $n$ variables over a field $k$ that is a prime element of the ring $k[x_1,\ldots,x_n]$, that is, it cannot be represented in the form $f=gh$ where $g$ and $h$ are non-constant polynomials with coefficients in $k$ (irreducibility over $k$). A polynomial is called absolutely irreducible if it is irreducible over the algebraic closure of its field of coefficients. The absolutely irreducible polynomials of a single variable are the polynomials of degree 1. In the case of several variables there are absolutely irreducible polynomials of arbitrarily high degree, for example, any polynomial of the form $f(x_1,\ldots,x_{n-1})+x_n$ is absolutely irreducible.

The polynomial ring $k[x_1,\ldots,x_n]$ is factorial (cf. Factorial ring): Any polynomial splits into a product of irreducibles and this factorization is unique up to constant factors. Over the field of real numbers any irreducible polynomial in a single variable is of degree 1 or 2 and a polynomial of degree 2 is irreducible if and only if its discriminant is negative. Over an arbitrary algebraic number field there are irreducible polynomials of arbitrarily high degree; for example, $x^n+px+p$, where $n>1$ and $p$ is a prime number, is irreducible in $\mathbf Q[x]$ by Eisenstein's criterion (see Algebraic equation).

Let $A$ be an integrally closed ring with field of fractions $k$ and let $f(x)\in A[x]$ be a polynomial in a single variable with leading coefficient 1. If $f(x)=g(x)h(x)$ in $k[x]$ and both $g(x)$ and $h(x)$ have leading coefficient 1, then $g(x),h(x)\in A[x]$ (Gauss' lemma).

Reduction criterion for irreducibility. Let $\sigma:A\to B$ be a homomorphism of integral domains. If $f(x)$ and $\sigma(f(x))$ have the same degree and if $\sigma(f(x))$ is irreducible over the field of fractions of $B$, then there is no factorization $f(x)=g(x)h(x)$ where $g(x),h(x)\in A[x]$ and $g(x)$ and $h(x)$ are not constant. For example, a polynomial $f(x)\in\mathbf Z[x]$ with leading coefficient 1 is prime in $\mathbf Z[x]$ (hence irreducible in $\mathbf Q[x]$) if for some prime $p$ the polynomial $\sigma(f(x))$ obtained from $f(x)$ by reducing the coefficients modulo $p$ is irreducible.

References

[1]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2]	S. Lang, "Algebra" , Addison-Wesley (1974)
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1–2 , Springer (1975)

Comments

A factorial ring is also known as a unique factorization domain (UFD).