Editing Field norm (section)

==Properties of the norm==
Several properties of the norm function hold for any finite extension.<ref>{{harvnb|Roman|2006|p=151}}</ref><ref name=":0" />

=== Group homomorphism ===
The norm '''N'''{{sub|''L''/''K''}} : ''L''* → ''K''* is a [[group homomorphism]] from the multiplicative group of ''L'' to the multiplicative group of ''K'', that is
:<math>\operatorname{N}_{L/K}(\alpha \beta) = \operatorname{N}_{L/K}(\alpha) \operatorname{N}_{L/K}(\beta) \text{ for all }\alpha, \beta \in L^*.</math>
Furthermore, if ''a'' in ''K'':
:<math>\operatorname{N}_{L/K}(a \alpha) = a^{[L:K]} \operatorname{N}_{L/K}(\alpha) \text{ for all }\alpha \in L.</math>

If ''a'' ∈ ''K'' then <math>\operatorname{N}_{L/K}(a) = a^{[L:K]}.</math>

=== Composition with field extensions ===
Additionally, the norm behaves well in [[tower of fields|towers of fields]]:

if ''M'' is a finite extension of ''L'', then the norm from ''M'' to ''K'' is just the composition of the norm from ''M'' to ''L'' with the norm from ''L'' to ''K'', i.e.
:<math>\operatorname{N}_{M/K}=\operatorname{N}_{L/K}\circ\operatorname{N}_{M/L}.</math>

=== Reduction of the norm ===
The norm of an element in an arbitrary field extension can be reduced to an easier computation if the degree of the field extension is already known. This is<blockquote><math>N_{L/K}(\alpha) = N_{K(\alpha)/K}(\alpha)^{[L:K(\alpha)]}</math><ref name=":0">{{Cite book|last=Oggier|url=http://www1.spms.ntu.edu.sg/~frederique/ANT10.pdf|title=Introduction to Algebraic Number Theory|pages=15|access-date=2020-03-28|archive-date=2014-10-23|archive-url=https://web.archive.org/web/20141023023935/http://www1.spms.ntu.edu.sg/~frederique/ANT10.pdf|url-status=dead}}</ref></blockquote>For example, for <math>\alpha = \sqrt{2}</math> in the field extension <math>L = \mathbb{Q}(\sqrt{2},\zeta_3), K =\mathbb{Q}</math>, the norm of <math>\alpha</math> is<blockquote><math>\begin{align}
N_{\mathbb{Q}(\sqrt{2},\zeta_3)/\mathbb{Q}}(\sqrt{2}) &= N_{\mathbb{Q}(\sqrt{2})/\mathbb{Q}}(\sqrt{2})^{[\mathbb{Q}(\sqrt{2},\zeta_3):\mathbb{Q}(\sqrt{2})]}\\
&= (-2)^{2}\\
&= 4
\end{align}</math></blockquote>since the degree of the field extension <math>L/K(\alpha)</math> is <math>2</math>.

=== Detection of units ===
For <math>\mathcal{O}_K</math> the [[ring of integers]] of an [[algebraic number field]] <math>K</math>, an element <math>\alpha \in \mathcal{O}_K</math> is a unit if and only if <math>N_{K/\mathbb{Q}}(\alpha) = \pm 1</math>.

For instance
:<math>N_{\mathbb{Q}(\zeta_3)/\mathbb{Q}}(\zeta_3) = 1</math>
where
:<math>\zeta_3^3 = 1</math>.

Thus, any number field <math>K</math> whose ring of integers <math>\mathcal{O}_K</math> contains <math>\zeta_3</math> has it as a unit.