# Splitting field of a polynomial

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 $\operatorname{GF}(p)\subset\operatorname{GF}(q)$.

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

#### References

 [a1] I. Stewart, "Galois theory" , Chapman & Hall (1979) [b1] Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) Template:ISBN
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=53040
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article