TITLE: Can the norm of a non-algebraic integer be an integer?
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: Hint $ $ Over $\,\Bbb Q,\,$ a quadratic number is an algebraic integer iff its $\,\color{#0a0}{\rm norm}\,$ and $\,\rm\color{#c00}{trace}$ are integers, i.e. iff its monic minimal polynomial has integer coefficients. So your question reduces to finding an irreducible polynomial $\,x^2\! + \color{#c00}b\,x + \color{#0a0}c\in\Bbb Q[x] $ with $\,\color{#c00}{b\not\in \Bbb Z},\ \color{#0a0}{c\in \Bbb Z},\,$ which is quite easy.
Remark $\ $ The same argument works over any integrally-closed domain $\,D,\,$ since then monic minimal polynomials of $D$-integral eements must have coefficients in $\,D\,$  (a property that is equivalent to $\,D\,$ being integrally-closed).