For which polynomial h, C[x] is separable over k+(h)?

25 January 2021 1 6K Report

Let h belong to C[x] (C = complex numbers). Denote by R_h the C-subalgebra of C[x] of the form C+(h), where (h) is the ideal of C[x] generated by h.

Claim: C[x] is separable over R_h iff h is linear (= of degree 1).

Question: Could one prove or refute this claim? I have not succeeded either prove or refute it. I suspect that the answer is not difficult, and my problem is that I lack the relevant algebraic geometry knowledge.

Example: h=x^2, so R_h=C+(x^2)=C[x^2,x^3], and C[x] is not separable over that R_h, see: https://mathoverflow.net/questions/371948/separability-of-mathbbcx-over-its-mathbbc-subalgebras?noredirect=1&lq=1

Thank you very much!

Taki Eddine Oussaeif

degree 1

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