Let L/K be a finite field extension and define the norm of an element as the product of each K-embedding evaluated at that element.
Can the norm of a non-algebraic integer be an integer?
I know that the norm of an algebraic integer is always an integer as it corresponds to the final term in the minimum polynomial, I was wondering if there was a converse to this- even under special conditions.
EDIT: Sorry yes an element which belongs to L.
Best Answer
The norm of $$ \frac35+\frac45i\in\Bbb Q[i] $$ is $$ \left(\frac35\right)^2+\left(\frac45\right)^2=1 $$ More generally, any primitive pythagorean triple $(a,b,c)$ can be used to construct elements in $\Bbb Q(i)$ of norm 1 which are not integers, just take $z=\frac ac+\frac bci$
Even more generally if $K$ is a quadratic field and $0\neq z\in K$ any, the quotient $z/\bar z$ has always norm 1.