Is there a name for an irreducible (separable, if you want) polynomial over a field such that adjoining one root of the polynomial splits the polynomial?
Such polynomials are discussed here for example.
2026-05-14 08:42:28.1778748148
Name for a polynomial where adjoining one root is Galois
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Your question is whether there’s an established name for these polynomials. As far as I know, there is no agreed-upon name, and it even seems to me that people haven’t bothered giving nonce-names to the concept either.
But now that I’m on my soapbox, why don’t I point out that if $L\supset K$ is a Galois extension and $\alpha$ is a primitive element (of which there are many many, of course), then the minimal $K$-polynomial for $\alpha$ has your property. If you want a ready-made infinite family of polynomials of this kind, you have the cyclotomic polynomials $\Phi_n$.