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

#### Comments

See also Galois theory; 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

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

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

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Splitting field of a polynomial.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=53040