Wow! Thanks for the detailed explanation. It is very clever, and I need to go review the use of Legendre symbols (it's been years since I last used it!). I take it that this has been a known result. Do you happen to know of a textbook/theorem name/paper? I'm very glad to know that it is indeed...
Thanks for all of the replies so far, and sorry for the confusion on \mathbb{Z}_p \backslash \{0\} . I didn't know how to make a backslash in TeX. The reason I'm asking this is because, in \mathbb{Z}_3 and in \mathbb{Z}_5 , I think all polynomials of the above form have a non-trivial root...
I've been doing some work and I keep running into polynomials of the following form:
P(x,y,z) = ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) \mod p
where a,b,c \in \mathbb{Z}_p/ \{0\} and d , e, f \in \mathbb{Z}_p . It would be great if I knew anything about the existence of roots of P ...
If you're not too experienced with proof writing, I recommend Elementary Analysis by Ross. It's a very gentle introduction to analysis. This book solely deals with analysis on the real numbers, and does not concern itself with metric spaces or anything. The problems are very friendly and I...
Homework Statement
I'm a bit confused at a single step in a proof. Let \phi \in L^1(\mathbb{R}) \cap C(\mathbb{R}^d) be a function such that for any \omega \in \mathbb{R}^d , \phi(\omega) = \psi(||\omega)|| . That is, the function depends solely on the norm of the vector input, so it is...
Homework Statement
Show that a finite field of p^n elements has exactly one subfield of p^m elements for each m that divides n. Homework Equations
If F \subset E \subset K are field extensions of F , then [K:F] = [E:F][K:F] . Also, a field extension over a finite field of p elements...