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I'd like to fine an example of field $K$ and elements $\alpha, \beta$ such that $\mathrm{char}(K) = p> 0 $, $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$.

This obviously can't work if $K$ is a finite field. So I need to find a non-finite $K$. The only ones that pop into my head are $\mathbb F_p(t)$, $\mathbb F_p(t^p)$ and $\overline{\mathbb F}_p$ for $t$ an indererminate, but I'm struggling to find an example.

Any hints would be greatly appreciated.

Thanks

  • 6
    $\mathbb{F}_p(t)$ will work. Try quadratic extensions.2012-05-08
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    Skillfully pick two irreducible quadratic polynomials in $F_3(t)[X]$ such that the corresponding quotients are not isomorphic.2012-05-08
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    @QiaochuYuan Would something simple like $K = \mathbb F_p(t)$, $\alpha = \sqrt{2}, \beta = i$ work?2012-05-08
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    Or would that only work in the case where $-1$ isn't a quadratic residue mod $p$?2012-05-08
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    @Jonathan: if $p$ is such that both of those are quadratic extensions, then both of those extensions are just $\mathbb{F}_{p^2}(t)$. You need to use $t$.2012-05-08
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    @QiaochuYuan I guess my problem is that I'm struggling to see what will stop the resulting extensions from being isomorphic.2012-05-08
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    @Jonathan: do you know how to show that two quadratic extensions of $\mathbb{Q}$ aren't isomorphic?2012-05-08
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    If $L_1 = \mathbb Q(\sqrt{a})$ and $L_2 = \mathbb Q(\sqrt{b})$ are two quadratic extensions with $a,b \in \mathbb Q$, then if $b$ has no square root in $L_1$ then the two aren't isomorphic (and likewise for $a$)2012-05-08
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    @Jonathan: yep. That's still true here.2012-05-08
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    @QiaochuYuan So taking something like $\alpha = \sqrt{t}$, $\beta = \sqrt{t^3}$ would work? Thanks for your help.2012-05-08
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    @Jonathan: no, those are isomorphic (since $\alpha = t \beta$). Use $t$ in one of the extensions but not the other!2012-05-08
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    @QiaochuYuan I think that's the worst mistake I've made since I first learned what a field is. I now understand what's going on, and withdraw sheepishly. Thanks for your help2012-05-08

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