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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:
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{{TEX|done}}{{MSC|12F}}
  
<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>
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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:
  
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" />.
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$$f=a_0(x-a_1)\dotsm(x-a_n)$$
  
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" />.
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and if $L=K(a_1,\dotsc,a_n)$ (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 $K$.
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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 $x^q-x$ over the prime subfield $\operatorname{GF}(p)\subset\operatorname{GF}(q)$.
  
  
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Stewart,  "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Stewart,  "Galois theory" , Chapman &amp; Hall  (1979)</TD></TR></table>
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====Comments====
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The splitting field of a polynomial is necessarily a [[normal extension]]: a finite degree normal extension is the splitting field of some polynomial.
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====References====
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<table><TR><TD valign="top">[b1]</TD> <TD valign="top">  Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) ISBN 048678147X </TD></TR></table>

Latest revision as of 13:09, 14 February 2020

2010 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]

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:

$$f=a_0(x-a_1)\dotsm(x-a_n)$$

and if $L=K(a_1,\dotsc,a_n)$ (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 $x^q-x$ over the prime subfield $\operatorname{GF}(p)\subset\operatorname{GF}(q)$.


Comments

See also Galois theory; Irreducible polynomial.

References

[a1] I. Stewart, "Galois theory" , Chapman & Hall (1979)

Comments

The splitting field of a polynomial is necessarily a normal extension: a finite degree normal extension is the splitting field of some polynomial.

References

[b1] Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) ISBN 048678147X
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