View source for Splitting field of a polynomial

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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868607.png" /></td> </tr></table>

and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086860/s0868608.png" /> (see [[Extension of a field|Extension 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" />.

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 field|finite 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" />.



====Comments====
See also [[Galois theory|Galois theory]]; [[Irreducible polynomial|Irreducible polynomial]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Stewart,   "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>