View source for Splitting field of a polynomial

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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:

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

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|Galois theory]]; [[Irreducible polynomial|Irreducible polynomial]].

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

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