When is an element of an extension field in the base field?
Given a finite field $mathbb{F}_p$, consider an extension of it; $mathbb{F}_{p^m}$. If I'm given $alpha in mathbb{F}_{p^m}$, then, if $alpha^p = alpha$, $alpha in mathbb{F}_{p}$.
Why is this true?
abstract-algebra field-theory finite-fields extension-field
asked Dec 9 at 12:24
the man
697715
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Given a finite field $mathbb{F}_p$, consider an extension of it; $mathbb{F}_{p^m}$. If I'm given $alpha in mathbb{F}_{p^m}$, then, if $alpha^p = alpha$, $alpha in mathbb{F}_{p}$.
Why is this true?
abstract-algebra field-theory finite-fields extension-field
asked Dec 9 at 12:24
the man
697715
add a comment |
Given a finite field $mathbb{F}_p$, consider an extension of it; $mathbb{F}_{p^m}$. If I'm given $alpha in mathbb{F}_{p^m}$, then, if $alpha^p = alpha$, $alpha in mathbb{F}_{p}$.
Why is this true?
abstract-algebra field-theory finite-fields extension-field
asked Dec 9 at 12:24
the man
697715
Given a finite field $mathbb{F}_p$, consider an extension of it; $mathbb{F}_{p^m}$. If I'm given $alpha in mathbb{F}_{p^m}$, then, if $alpha^p = alpha$, $alpha in mathbb{F}_{p}$.
Why is this true?
abstract-algebra field-theory finite-fields extension-field
abstract-algebra field-theory finite-fields extension-field
asked Dec 9 at 12:24
the man
697715
asked Dec 9 at 12:24
the man
697715
asked Dec 9 at 12:24
the man
697715
asked Dec 9 at 12:24
the man
697715
asked Dec 9 at 12:24
the man
697715
697715
add a comment |
add a comment |
1 Answer
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The equation $x^p-x=0$ has at most $p$ roots. But it has $p$ roots within $Bbb F_p$. So any root in an extension of $Bbb F_p$ already lies in $Bbb F_p$.
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
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1 Answer
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1 Answer
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The equation $x^p-x=0$ has at most $p$ roots. But it has $p$ roots within $Bbb F_p$. So any root in an extension of $Bbb F_p$ already lies in $Bbb F_p$.
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
add a comment |
The equation $x^p-x=0$ has at most $p$ roots. But it has $p$ roots within $Bbb F_p$. So any root in an extension of $Bbb F_p$ already lies in $Bbb F_p$.
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
add a comment |
The equation $x^p-x=0$ has at most $p$ roots. But it has $p$ roots within $Bbb F_p$. So any root in an extension of $Bbb F_p$ already lies in $Bbb F_p$.
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
The equation $x^p-x=0$ has at most $p$ roots. But it has $p$ roots within $Bbb F_p$. So any root in an extension of $Bbb F_p$ already lies in $Bbb F_p$.
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
answered Dec 9 at 12:33
Lord Shark the Unknown
101k958132
101k958132
add a comment |
add a comment |
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