More norms

Dear Prof.,

1) I noticed this pattern in some of the calculations, is this correct?

$N(c*(a^n))=c^c*(N(a))^n,$
where $a$ is a root of the minimal polynomial; $c$ and $n$ are constants.

2) I am following the solution of 2008 past paper question 2. Is it true that $a$ is an algebraic integer if its norm and trace are integers.

3) When distinguishing between two fields, $F$ and $F'$ with basis $B$ and $B'$, we check that if $\Delta(B)/\Delta(B')$ is in $(Q^*)^2$ or not. Does this simply mean if $\Delta(B)/\Delta(B')$ is square-free in the rational?
—————————————————————–

The general nature of your questions indicates you should review the course more thoroughly from the beginning. I will try to indicate just a few points where your questions are ill-formed and/or worrisome.

(1) When you say $c$ and $n$ are constants,’ it’s very unclear what this means. All of the quantities are constant’ in the usual sense that they are numbers. Also, when you say $a$ is a root of the minimal polynomial’ what minimal polynomial are you referring to?

One general fact about norms of numbers that follows immediately from the definitions (that is, you should know how to prove it easily at this point) is that

$N(ab)=N(a)N(b).$

In particular, $N(a^n)=N(a)^n$. Of course, $n$ should be a *natural number* here, that is, $n=0,1,2, \ldots$. When you refer to $c$ as a constant, you might mean that it is rational. In that case, if the norm is being taken inside a number field $F$ of degree $d$, then $N(c)=c^d$. Therefore, if $c$ is rational and $a$ is any element of $F$, then

$N(ca^n)=N(c)N(a)^n=c^dN(a)^n.$

Once again, you should have a very clear understanding of how this equality comes about starting from the definition. The worst possible approach to this material is to try to pick up on patterns of equalities that occur in the notes and then vaguely guess which variations on them are admissible. You may not be doing this, but this strategy is unfortunately common, so I’m giving a strong warning against it whenever I can. Note that in this discussion, whether $a$ is the root of some specific minimal polynomial or is irrelevant. Try to understand for yourself why that phrase occurred in your question. (Of course any algebraic number is a root of its *own* minimal polynomial.)

(2) This question makes sense. The fact is that the norm and trace of any algebraic integer are integers. However, the converse is false in general. On the other hand, if $a \in F$ and $F$ is a quadratic field, then $a$ is an algebraic integer if and only if it norm and trace are integers. Prove this last statement as an exercise.

(3) The word square-free’ for an integer $n$ means that $n$ is not divisible by the square of any prime. A rational number $n/m$ written in reduced form is square-free if both $n$ and $m$ are square-free integers. A rational number $x$ is a square if there is a rational number $y$ such that $x=y^2$. Be careful not to confuse these elementary concepts. Once again, the way you formulated the question is itself a bit worrisome. For example, the sentence that begins `Does this simply mean…’ is very bad in many ways.

Regarding the motivating problem, if $F$ and $F'$ are isomorphic, then for any choice of bases $B$ and $B'$, we would have that $\Delta(B)/\Delta(B')$ is a square. In particular, if $\Delta(B)/\Delta(B')$ is not a square for some bases $B$ and $B'$, then $F$ and $F'$ are *not* isomorphic. One should be aware that the converse is false: $\Delta(B)/\Delta(B')$ being a square certainly does not imply that $F$ and $F'$ are isomorphic. Examples illustrating all this are contained in the notes.