number field

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- Tags: - #NT/algebraic - Refs: - #todo/add-references - Links: - CM field

number field

Any finite extension \(K/{\mathbf{Q}}\), so \([K: {\mathbf{Q}}] < \infty\).

Can take ring of integers \({\mathcal{O}}_K\) as the integral closure of \({\mathbf{Z}}\) in \(K\), or equivalently the algebraic integers in \(K\). Note \(\operatorname{ff}({\mathcal{O}}_K) = K\) and \({\mathcal{O}}_K\) is a Dedekind domain of Krull dimension one.

Can always find a power basis by the primitive element theorem.

Can define trace and norm of \(x\) as the trace and determinant of the map \(m_x: a\mapsto ax\).

Has archimedean places and nonarchimedean places.

See also ramification, inertia group

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algebraic in Unsorted/number field

primitive element theorem in Unsorted/number field
unramified

Relation to ramification in number fields:
section conjecture

An instance of the anabelian philosophy, Grothendieck’s section conjecture: for a ‘nice’ curve \(X\) over a number field \(F\), the rational points are in bijection with the sections of the exact sequence \begin{align*} 1 \rightarrow \pi_1^\text{ét}(X_{\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu}) \rightarrow \pi_1^ \text{ét}(X) \rightarrow { \mathsf{Gal}} (F^s/F) \rightarrow 1 \end{align*}
ring of integers

number field
perfect field

\({\mathbf{Q}}, {\mathbf{R}}, {\mathbf{C}}\), any number field
Riemann Hypothesis

More generally, the Dedekind zeta function \(\zeta_K(s)\) of a number field \(K\), the conjecture is that if \(s\) is a zero with \(\Re(s) \in (0, 1)\), then \(\Re(s) = {1\over 2}\).

#NT/algebraic #todo/add-references