Namespaces
Variants
Actions

Splitting field of a polynomial

From Encyclopedia of Mathematics
Jump to: navigation, search

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:

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

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 $x^q-x$ 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://www.encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=34455
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article