Let $F/K$ be a normal finite extension where $F=F_1 \times F_2$ for subfields $F_1,F_2$ of $F$ where $F_1\ne F$ . Suppose $w$ is a normal basis element in $F/K$ for any $w\in F$ for which $Tr_{F/F_2}(w)$ is a normal basis element in $F_2/K$ . Then is it true that $K$ has prime characteristic ?

Let $F/K$ be a normal finite extension where $F=F_1 \times F_2$ for subfields $F_1,F_2$ of $F$ where $F_1\ne F$ . Suppose $K$ has prime characteristic . Then is it true that $w$ is a normal basis element in $F/K$ for any $w\in F$ for which $Tr_{F/F_2}(w)$ is a normal basis element in $F_2/K$ ? ...

Suppose that $G$ is a signed graph (a graph in which each edge has a positive of negative sign) Define Adjacency matrix of $G$ as before except that we have $+1$ for positive edges and $-1$ Prove that the eigen values of Adjacency matrix will not change if you multiply some rows and columns by...