Abstract Algebra

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Finite Fields: Squares and Sums of Two Squares

ABSALG-DDRUVY

Let $F$ be a finite field of characteristic $p\ge 2$.

Which of the following are always true?

Select ALL that apply.

A

Exactly half of the elements of $F$ are squares when $p=2$ and
exactly half the elements of $F^\ast=F\setminus\{0\}$ are squares when $p\ge 3$.

B

The map $\varphi:F\rightarrow F$ given by $\varphi(a)=a^2$ is a ring homomorphism.

C

If $A$ and $B$ are subsets of $F$ such that $|A|+|B|$ is strictly greater than $|F|$, then $F=\{a+b\mid a\in A, b\in B\}$.

D

Every element of $F$ is a sum of two squares (the squares are not necessarily distinct and may be zero).