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 $L$ 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: | 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: | ||
| − | $$f=a_0(x-a_1)\ | + | $$f=a_0(x-a_1)\dotsm(x-a_n)$$ |
| − | and if $L=K(a_1,\ | + | 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)$. | 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)$. | ||
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See also [[Galois theory|Galois theory]]; [[Irreducible polynomial|Irreducible polynomial]]. | See also [[Galois theory|Galois theory]]; [[Irreducible polynomial|Irreducible polynomial]]. | ||
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The splitting field of a polynomial is necessarily a [[normal extension]]: a finite degree normal extension is the splitting field of some 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==== | ====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> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> I. Stewart, "Galois theory" , Chapman & Hall (1979)</TD></TR> | ||
| + | <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 14:06, 20 March 2023
2020 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) ISBN 048678147X |
Splitting field of a polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=36927