Difference between revisions of "Splitting field of a polynomial"
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| − | The smallest field containing all roots of that polynomial. More exactly, an extension | + | {{TEX|done}} |
| + | 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 | + | and if $L=K(a_1,\ldots,a_n)$ (see [[Extension of a field|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 | + | 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|finite field]] $\operatorname{GF}(q)$, where $q=p^n$, is the splitting field of the polynomial $x^q-x$ over the prime subfield $\operatorname{GF}(p)\subset\operatorname{GF}(q)$. |
Revision as of 07:02, 15 July 2014
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) |
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=32434
Splitting field of a polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=32434
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article