If $A$ is a fractional ideal of $\mathbb{Q}(\sqrt{-m})$, then $A^{1+\sigma}$ is an ideal in $\mathbb{Q}$. I'm stuck on this little detail in Washington's intro to cyclotomic fields.
Let $M = \mathbb{Q}(\sqrt{-m})$, and $A$ be a fractional ideal of $M$.  With $Gal(M/\mathbb{Q})= \{1,\sigma\}$, Washington says that $A^{1 + \sigma} = A\cdot A^{\sigma}$ is an ideal in $\mathbb{Q}$.  I'm a little confused, but I think what he means is that $A^{1+\sigma} = q\mathcal{O}_M$ for some $q \in \mathbb{Q}$ (the point of this is to show that $A^{1+\sigma}$ is principal).
With $A = \alpha I$ for some ideal $I$ in $\mathcal{O}_M$ and $\alpha \in M$, we have
\begin{align*}
A^{1+\sigma} = \alpha\overline{\alpha}(I\cdot I^{\sigma})
\end{align*}
and $\alpha\overline{\alpha} \in \mathbb{Q}$.  I guess it then comes down to showing that $I\cdot I^{\sigma}$ is principal (not sure if this would even be true).
Is this the right interpretation?
 A: EDIT after discussion with the OP.
I keep your hypotheses, with the additional notations $I,J$,etc. for integral ideals, i.e. ideals of the ring of integers $S$ of $M$. Write also $N$ for the usual norm map of $M/\mathbf Q$. The absolute norm $N(I)$ (there will be no confusion with the notation) is classically defined as card $(S/I)$ up to a sign. As $\pm 1$ are the only units of $\mathbf Z$, we may as well view $N(I)$ as an ideal of $\mathbf Z$, and extend it to S by considering the ideal $N(I).S$. Define also the ideal $\nu(I)=I.\sigma(I)$ of S. To stress the difference, $N(I)$ (resp. $\nu(I)$) is sometimes called the arithmetic (resp. algebraic) norm of I. You want to show that $\nu(I)=N(I).S$, hence in particular is principal. Proof. Since S is a Dedekind ring, the question is brought back to the case where $I$ is a prime ideal $P$ of $S$. Let $p$ be the prime number under $P$. There are 3 types of decomposition of $p$ in $S$ : $p$ is inert, i.e. $pS$ is prime; $p$ splits, i.e. $pS=P.\sigma(P)$; $p$ is (totally) ramified, i.e. $pS=P^2$ (you seem to have forgotten this last possibility in your reply to @reuns). The norm calculations relative to these distinct cases can be synthetized in the single formula $N(P)=p^f$, where $f=f(P/p)$ is the inertia index of $P$ over $p$. This also gives $N(P).S$, which can be checked to be equal to $P.\sigma(P)$ using the decomposition of $p$ in $S$.
I gave all the (ponderous) details because this approach can be directly extended to an arbitrary finite Galois extension of number fields $M/K$ with Galois group $G$. (NB: this is not merely generalization for the sake of generalization. The norm of ideals is an essential tool in class field theory.) Let $R,S$ resp. be the ring of integers of $K,M$. The point is the definition of the arithmetic norm $N(I)$. There are many equivalent definitions, but the equivalence is sometimes not quite straightforward. See e.g. https://mathoverflow.net/questions/9235/a-problem-on-algebraic-number-theory-norm-of-ideals. Following D. Marcus, "Number Fields", chapter 3, define $\nu(I)= \prod \sigma I$, for all $\sigma \in G$, and $N(I)=R \cap \nu(I)$. Then it can be shown along our previous line (but of course in a more complicated way) that $N(I).S=\nu(I)$. For details, see op. cit. chapter 3, exercise 14.
