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Jul
7
comment If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$
That result is one of my theorems yes, and Zev's comment answers my question (thank you Zev). But is knowing both contain all the roots of that poly enough without knowing it's separable? Not that it's particularly important, anyways thanks for the help.
Jul
7
comment If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$
It seems intuitively obvious, but how do we know all the roots of $x^q - x$ are distinct?
Jul
7
revised If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$
added 42 characters in body
Jul
7
comment If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$
Sorry I should have mentioned, I have no Galois Theory at my disposal. Group Theory up through Sylow Theorems, and Field Theory through splitting fields and general results on the classification of finite fields. Also undergrad ring theory, but that's probably not important.
Jul
7
asked If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$
Jul
6
accepted A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
Jul
4
accepted Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
Jul
4
accepted Trying to sort the coefficients of the polynomial $(z-a)(z-b)(z-c)…(z-n)$ into a vector
Jul
4
accepted Pairwise non-integral numbers
Jul
4
comment A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
@Arturo Magidin: you're right, thanks for pointing that out.
Jul
4
revised A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
added 17 characters in body
Jul
4
revised A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
added 12 characters in body
Jul
4
comment A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
Ah yes sorry forgot to say that, $f(x)$ is irreducible in $F[x]$. Normal means if it is irreducible in $F[x]$ and has one root in $K[x]$ then it has all roots in $K[x]$
Jul
4
asked A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
Jul
3
comment Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
@Dylan Moreland: Ok cool, thanks for the help. =]
Jul
3
comment Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
ohh nice! but we still have $\mathbb{Q}(\sqrt[3]{2})\cong \mathbb{Q}(\sqrt[3]{2}w)\cong \mathbb{Q}[x]/x^3 -2$, where $w$ is a third root of unity, correct?
Jul
3
comment Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
Ok interesting, so is there a canonical example of when the quotient field does not contain all the roots of the polynomial?
Jul
3
comment Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
I thought forming the quotient field adjoined just a single root? And that sometimes you would get lucky and the remaining roots could be formed within the quotient field as well, but that sometimes they couldn't, is this not correct?
Jul
3
revised Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
added 123 characters in body
Jul
3
comment Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
oh shoot I must have miscaculated in my quotient field, ok cool, thanks!