# Difference between revisions of "Splitting field of a polynomial"

(Importing text file) |
m (dots) |
||

(5 intermediate revisions by 2 users not shown) | |||

Line 1: | Line 1: | ||

− | + | {{TEX|done}}{{MSC|12F}} | |

− | + | 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: | |

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

− | Examples. The field of complex numbers | + | 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 $\operatorname{GF}(p)\subset\operatorname{GF}(q)$. | ||

Line 14: | Line 16: | ||

====References==== | ====References==== | ||

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

+ | |||

+ | ====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==== | ||

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

## Contents

#### 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