Namespaces
Variants
Actions

Difference between revisions of "Irreducible polynomial"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526201.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526202.png" /> variables over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526203.png" /> that is a prime element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526204.png" />, that is, it cannot be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526205.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526207.png" /> are non-constant polynomials with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526208.png" /> (irreducibility over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i0526209.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262010.png" /> is absolutely irreducible.
+
{{TEX|done}}
 +
A polynomial $f=f(x_1,\ldots,x_n)$ in $n$ variables over a field $k$ that is an irreducible element of the [[Ring of polynomials|polynomial 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262011.png" /> is factorial (cf. [[Factorial ring|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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262014.png" /> is a prime number, is irreducible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262015.png" /> by Eisenstein's criterion (see [[Algebraic equation|Algebraic equation]]).
+
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|Algebraic equation]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262016.png" /> be an integrally closed ring with field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262017.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262018.png" /> be a polynomial in a single variable with leading coefficient 1. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262020.png" /> and both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262022.png" /> have leading coefficient 1, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262023.png" /> (Gauss' lemma).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262024.png" /> be a homomorphism of integral domains. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262026.png" /> have the same degree and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262027.png" /> is irreducible over the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262028.png" />, then there is no factorization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262029.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262032.png" /> are not constant. For example, a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262033.png" /> with leading coefficient 1 is prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262034.png" /> (hence irreducible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262035.png" />) if for some prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262036.png" /> the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262037.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262038.png" /> by reducing the coefficients modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052620/i05262039.png" /> is irreducible.
+
Reduction criterion for irreducibility. Let $\sigma\colon A\to B$ 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====
 
====References====
Line 14: Line 15:
 
====Comments====
 
====Comments====
 
A factorial ring is also known as a unique factorization domain (UFD).
 
A factorial ring is also known as a unique factorization domain (UFD).
 +
 +
[[Category:Field theory and polynomials]]

Latest revision as of 15:26, 30 December 2018

A polynomial $f=f(x_1,\ldots,x_n)$ in $n$ variables over a field $k$ that is an irreducible element of the polynomial 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 $\sigma\colon A\to B$ 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).

How to Cite This Entry:
Irreducible polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_polynomial&oldid=12333
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article