Adding an imaginary unit to a finite field(johndcook.com)
johndcook.com
Adding an imaginary unit to a finite field
https://www.johndcook.com/blog/2025/11/16/finite-field-i/
6 コメント
Why is that? My guess would be that you could adjoin an i all the time to the p^n field and get the p^2n field, as long as you had p = 4k + 3. But that's admittedly based on approximately zero thinking.
EDIT: Looking things up indicates that if n is even, there's already a square root of -1 in the field, so we can't add another. So now I believe the 1/4 of the time thing you mentioned, and can't see how that's wrong.
EDIT: Looking things up indicates that if n is even, there's already a square root of -1 in the field, so we can't add another. So now I believe the 1/4 of the time thing you mentioned, and can't see how that's wrong.
Spitballing here, but I suspect it's a density thing. If you are considering all prime powers up to some bound N, then the density of prime powers (edit: of size p^n with n > 1) approaches 0 as N tends to infinity. So rather than things being 1/4 like our intuition says, it should unintuitively be 1/2. I haven't given this much thought, but I suspect this based on checking some examples in Sage.
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Of course, considering finite fields of prime power order, one might leap to the conclusion "a quarter of the time". One can adjoin "i" for prime powers p^n for half the primes and odd n.
Alas, this be wrong, for an amusing reason.