= Product formula
{wiki}
For a nonzero element $x$ of a <number field> $L$, normalized absolute values satisfy
$$
\prod_{v\in M_L}|x|_v^{[L_v:\mathbb Q_v]}=1.
$$
For an algebraic integer $x$, factor the principal ideal $(x)$ into prime ideals. Its ideal norm is both the product of the finite-place contributions with inverse exponent and the absolute value of the product of its Archimedean conjugates. Equating the two expressions proves the formula for algebraic integers, and writing an arbitrary $x\in L^\times$ as a quotient proves the general case.
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