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In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking values in the i …

It is quite easy to show that if $A$ is a Dedekind domain and $\mathfrak{p}\in \operatorname{Spec} A$, then if $A_{\mathfrak{p}}$ is the completion of $A$ at $\mathfrak{p}$ and $A_{(\mathfrak{p})}=(A\ …

I am referring to the very useful K. Conrad's notes on the conductor ideal of an order in a Dedekind domain: https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf $\DeclareMathOperator\Cl{Cl} …

Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage which looks …

Let $L/K/F$ be a tower of separable finite field extensions and let $x\in L$ such that $L=K(x)$. Under which conditions is it possible to choose $x$ such that $N_{L/K}(x)$ is again a primitive generat …

In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field …