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−  The smallest field containing all roots of that polynomial. More exactly, an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868601.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868602.png" /> is called the splitting field of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868603.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868604.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868605.png" /> decomposes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868606.png" /> into linear factors:  +  {{TEXdone}} 
 +  The smallest field containing all roots of that polynomial. More exactly, an extension $L$ of a field $K$ is called the splitting field of a polynomial $f$ over the field $K$ if $f$ decomposes over $L$ into linear factors: 
   
−  <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;textalign:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868607.png" /></td> </tr></table>
 +  $$f=a_0(xa_1)\ldots(xa_n)$$ 
   
−  and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868608.png" /> (see [[Extension of a fieldExtension of a field]]). A splitting field exists for any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868609.png" />, and it is defined uniquely up to an isomorphism that is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686010.png" />. It follows from the definition that a splitting field is a finite algebraic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686011.png" />.  +  and if $L=K(a_1,\ldots,a_n)$ (see [[Extension of a fieldExtension of a field]]). A splitting field exists for any polynomial $f\in K[x]$, and it is defined uniquely up to an isomorphism that is the identity on $K$. It follows from the definition that a splitting field is a finite algebraic extension of $K$. 
   
−  Examples. The field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686012.png" /> serves as the splitting field of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686013.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686014.png" /> of real numbers. Any [[Finite fieldfinite field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686016.png" />, is the splitting field of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686017.png" /> over the prime subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s08686018.png" />.  +  Examples. The field of complex numbers $\mathbf C$ serves as the splitting field of the polynomial $x^2+1$ over the field $\mathbf R$ of real numbers. Any [[Finite fieldfinite field]] $\operatorname{GF}(q)$, where $q=p^n$, is the splitting field of the polynomial $x^qx$ over the prime subfield $\operatorname{GF}(p)\subset\operatorname{GF}(q)$. 
   
   
Revision as of 07:02, 15 July 2014
The smallest field containing all roots of that polynomial. More exactly, an extension $L$ of a field $K$ is called the splitting field of a polynomial $f$ over the field $K$ if $f$ decomposes over $L$ into linear factors:
$$f=a_0(xa_1)\ldots(xa_n)$$
and if $L=K(a_1,\ldots,a_n)$ (see Extension of a field). A splitting field exists for any polynomial $f\in K[x]$, and it is defined uniquely up to an isomorphism that is the identity on $K$. It follows from the definition that a splitting field is a finite algebraic extension of $K$.
Examples. The field of complex numbers $\mathbf C$ serves as the splitting field of the polynomial $x^2+1$ over the field $\mathbf R$ of real numbers. Any finite field $\operatorname{GF}(q)$, where $q=p^n$, is the splitting field of the polynomial $x^qx$ over the prime subfield $\operatorname{GF}(p)\subset\operatorname{GF}(q)$.
See also Galois theory; Irreducible polynomial.
References
[a1]  I. Stewart, "Galois theory" , Chapman & Hall (1979) 
How to Cite This Entry:
Splitting field of a polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=16268
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics  ISBN 1402006098.
See original article