I'm learning finite fields. The equation $ax^2 + bx + c = 0$ can be solved in the reals trivially. If $a, b, c, x$ are elements of a finite field, is there a procedure to solve it?
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It depends upon the characteristic of the field. – José Carlos Santos Jan 16 '22 at 22:46
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@JoséCarlosSantos Could you elaborate? I'm particularly interested in $GF(2^{4096})$, but would appreciate general explanation. – SRobertJames Jan 16 '22 at 22:47
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Then you are interested in working over a field whose characteristic is $2$. Take a look here. – José Carlos Santos Jan 16 '22 at 22:49
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For finite fields in particular there is more definitve theory than what is given in the thread José linked to. For me personally it has always at the level of a FAQ. The theory is here. I should probably add an explanation of the use of the so called half trace. Probably the thread José linked to is a good place to do that. – Jyrki Lahtonen Feb 27 '22 at 18:58