Instead of having a single polynomial-quotient (aka “rupture”) step $\mathbb{F}_p[X]/(P)$ with $P \in \mathbb{F}_p[X]$ irreducible, you can also construct finite fields in several steps, i.e., construct first $\mathbb{F}_q = \mathbb{F}_p[X]/(P)$ with $P \in \mathbb{F}_p[X]$ irreducible and then the desired field as $\mathbb{F}_q[Y]/(Q)$ with $Q \in \mathbb{F}_q[Y]$ irreducible. (Even more generally, you can quotient $\mathbb{F}_p[X,Y]$ by a maximal ideal, though whether this is really different depends on what you mean by “different”, as a Gröbner basis for the lexigraphic ordering on monomials lets you write the latter as the former.) And of course you can have many such “rupture” steps.

This is, in fact, what happens for Conway's construction: $\mathbb{F}_{2^{2^{r+1}}}$ is constructed as the quotient of $\mathbb{F}_{2^{2^r}}$ by the Artin-Schreier polynomial $X^2 + X + c$ where $c$ is the element represented by $2^{2^r-1}$ (which is also the product, of the elements represented by $2^{2^i}$ for $0\leq i\leq r-1$), after what the element given by $a_1 X + a_0$ is represented by the integer $a_1 \cdot 2^{2^r} + a_0$, i.e., by juxtaposing the binary representations. So Conway's construction can be seen as an iteration of the quadratic rupture step given by an Artin-Schreier polynomial (constructing the A-S root of the “smallest” element $c$ which doesn't already have one).

Various people have given thought to the question of constructing the lattice of all finite fields (of a given characteristic) “uniformly” or “standardly”, or at least in a computably efficient way, by piling up rupture steps. For more about this, I can refer to Édouard Rousseau's 2021 PhD thesis “Arithmétique efficace des extensions de corps finis” [despite the title in French, most of the content is in English] and/or Frank Lübeck's paper “Standard Generators of Finite Fields and their Cyclic Subgroups”. The main issue is how to choose the rupture polynomials in a coherent way, and this can be seen as a generalization of Conway's construction.

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