Relation between finite fields $F_{p^2}$ and $F_p(omega)$.
$begingroup$
Let $F_p$ be a finite field having $p$ elements(for some prime $p$), not having primitive
$3$rd root of unit say $omega.$ Then can i say that field $F_p(omega)$ and $F_{p^2}$ are isomorphic.
I tried as follows: As both of fields $F_{p^{2}}$ and $F_p(omega)$ are of same dimension $2$ over the field $F_p$ and hence same number of elements. Now any two finite field with same number of elements are isomorphic.
Please suggest me. Thanks in advance.
finite-fields
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
$endgroup$
add a comment |
$begingroup$
Let $F_p$ be a finite field having $p$ elements(for some prime $p$), not having primitive
$3$rd root of unit say $omega.$ Then can i say that field $F_p(omega)$ and $F_{p^2}$ are isomorphic.
I tried as follows: As both of fields $F_{p^{2}}$ and $F_p(omega)$ are of same dimension $2$ over the field $F_p$ and hence same number of elements. Now any two finite field with same number of elements are isomorphic.
Please suggest me. Thanks in advance.
finite-fields
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
$endgroup$
1
$begingroup$
Yes, it is correct. Note that $omega$ is the root $x^2+x+1$, thus having order $2$.
$endgroup$
– Berci
Dec 9 '18 at 10:43
$begingroup$
Okh .... thanks .....
$endgroup$
– neelkanth
Dec 9 '18 at 11:02
add a comment |
$begingroup$
Let $F_p$ be a finite field having $p$ elements(for some prime $p$), not having primitive
$3$rd root of unit say $omega.$ Then can i say that field $F_p(omega)$ and $F_{p^2}$ are isomorphic.
I tried as follows: As both of fields $F_{p^{2}}$ and $F_p(omega)$ are of same dimension $2$ over the field $F_p$ and hence same number of elements. Now any two finite field with same number of elements are isomorphic.
Please suggest me. Thanks in advance.
finite-fields
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
$endgroup$
Let $F_p$ be a finite field having $p$ elements(for some prime $p$), not having primitive
$3$rd root of unit say $omega.$ Then can i say that field $F_p(omega)$ and $F_{p^2}$ are isomorphic.
I tried as follows: As both of fields $F_{p^{2}}$ and $F_p(omega)$ are of same dimension $2$ over the field $F_p$ and hence same number of elements. Now any two finite field with same number of elements are isomorphic.
Please suggest me. Thanks in advance.
finite-fields
finite-fields
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
edited Dec 9 '18 at 9:34
neelkanth
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
asked Dec 9 '18 at 8:56
neelkanthneelkanth
2,20921028
2,20921028
1
$begingroup$
Yes, it is correct. Note that $omega$ is the root $x^2+x+1$, thus having order $2$.
$endgroup$
– Berci
Dec 9 '18 at 10:43
$begingroup$
Okh .... thanks .....
$endgroup$
– neelkanth
Dec 9 '18 at 11:02
add a comment |
1
$begingroup$
Yes, it is correct. Note that $omega$ is the root $x^2+x+1$, thus having order $2$.
$endgroup$
– Berci
Dec 9 '18 at 10:43
$begingroup$
Okh .... thanks .....
$endgroup$
– neelkanth
Dec 9 '18 at 11:02
1
1
$begingroup$
Yes, it is correct. Note that $omega$ is the root $x^2+x+1$, thus having order $2$.
$endgroup$
– Berci
Dec 9 '18 at 10:43
$begingroup$
Yes, it is correct. Note that $omega$ is the root $x^2+x+1$, thus having order $2$.
$endgroup$
– Berci
Dec 9 '18 at 10:43
$begingroup$
Okh .... thanks .....
$endgroup$
– neelkanth
Dec 9 '18 at 11:02
$begingroup$
Okh .... thanks .....
$endgroup$
– neelkanth
Dec 9 '18 at 11:02
add a comment |
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$begingroup$
Yes, it is correct. Note that $omega$ is the root $x^2+x+1$, thus having order $2$.
$endgroup$
– Berci
Dec 9 '18 at 10:43
$begingroup$
Okh .... thanks .....
$endgroup$
– neelkanth
Dec 9 '18 at 11:02