The Number of Elements in a Finite Field is a Power of a Prime Number
Problem 726
Let $\F$ be a finite field of characteristic $p$.
Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$.
Add to solve later Tagged: field of characteristic p
Add to solve later Each Element in a Finite Field is the Sum of Two Squares
Problem 511
Let $F$ be a finite field.
Prove that each element in the field $F$ is the sum of two squares in $F$.
Add to solve later In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.
Problem 91
Show that the matrix $A=\begin{bmatrix}
1 & \alpha\\
0& 1
\end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$.
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