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I know field extensions of a finite field $\mathbb{F}_p$ for $p$ prime are represented as a quotient group over irreducible polynomials.

For example $\mathbb{F}_{2^2}\cong\mathbb{F}_2[x]/(x^2+x+1)=\{0,1,\alpha,\alpha+1\}$ where $\alpha$ is a root of $x^2+x+1$. Is there a way to label the elements of $\mathbb{F}_{2^2}$ as $\{0,1,2,3\}$ and work modulo some number? I haven't been able to make a connection.

Shean
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    Not sure what you are asking. $\mathbb Z/4\mathbb Z$ is not a field. Of course, you can name the elements of $\mathbb F_{2^2}$ anything you like, including ${0,1,2,3}$ but the field operations will not be straightforward. Note, of course, that using, e.g. $2$ as a symbol would be extremely confusing. You are much better off with ${0,1,\alpha, \alpha^2}$. – lulu Dec 11 '22 at 18:50
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    Multiplication modulo $4$ is noninvertible. E.g. $2\times2=0$ where $2\neq0$.

    Working $\mod p$ gives us a field iff. $p$ is prime. Otherwise, $\Bbb Z/p\Bbb Z$ is merely a ring.

     – FShrike Dec 11 '22 at 18:51
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    As @lulu says, the given additive and multiplicative structure on ${0,1,\cdots,p^n-1}$ have nothing to do with the additive and multiplicative structure on $\Bbb F_{p^n}$. You might as well call the elements of $\Bbb F_4$ “red, blue, green, yellow” as “zero, one, two, three”. – Lubin Dec 11 '22 at 19:36

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$\mathbb F_p=\mathbb Z/p\mathbb Z$ but, for example and for using the smallest extension, $\mathbb F_{p^2}\ne\mathbb Z/p^2\mathbb Z$. In particular $p$ as element of $\mathbb Z/p^2\mathbb Z$ is non-invertible but $p$ as non-zero element of the field $\mathbb F_{p^2}$ should be invertible. This would be enough to make the difference.

However an stronger argument is that the complement of $\mathbb F_p$ in its extension $\mathbb F_{p^2}$ are not simple classes modulo $p^2$ of integers but very strange (we could say of unimaginable nature) mathematical objects. If the number $\sqrt{-1}$ was "imaginary" for ancient people, an object in an algebraic extension of $\mathbb F_p$ would be "superimaginary" for us.

Piquito
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In the field $\mathbb{F}_4$, $1 + 1 = 0$. That field has characteristic $2$. In the integers modulo $4$ you have $1 + 1 = 2 \ne 0$.

Ethan Bolker
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