Editing Field norm (section)

==Formal definition==
Let ''K'' be a [[field (mathematics)|field]] and ''L'' a [[Degree of a field extension|finite]] [[field extension|extension]] (and hence an [[algebraic extension]]) of ''K''.

The field ''L'' is then a [[dimension (vector space)|finite-dimensional]] [[vector space]] over ''K''.

Multiplication by ''α'', an element of ''L'',
:<math>m_\alpha\colon L\to L</math>
:<math>m_\alpha (x) = \alpha x</math>,
is a ''K''-[[linear transformation]] of this vector space into itself.

The '''norm''', '''N'''<sub>''L''/''K''</sub>(''α''), is defined as the [[determinant]] of this linear transformation.<ref name=ROT940>{{harvnb|Rotman|2002|loc=p. 940}}</ref>

If ''L''/''K'' is a [[Galois extension]], one may compute the norm of ''α'' &isin; ''L'' as the product of all the [[Galois conjugate]]s of ''α'':

:<math>\operatorname{N}_{L/K}(\alpha)=\prod_{\sigma\in\operatorname{Gal}(L/K)} \sigma(\alpha),</math>

where Gal(''L''/''K'') denotes the [[Galois group]] of ''L''/''K''.<ref>{{harvnb|Rotman|2002|loc=p. 943}}</ref> (Note that there may be a repetition in the terms of the product.)

For a general field extension ''L''/''K'', and nonzero ''α'' in ''L'', let ''σ''{{sub|1}}(''α''), ..., σ{{sub|''n''}}(''α'') be the [[root of a polynomial|roots]] of the [[minimal polynomial (field theory)|minimal polynomial]] of ''α'' over ''K''  (roots listed with multiplicity and lying in some extension field of ''L''); then
:<math>\operatorname{N}_{L/K}(\alpha)=\left (\prod_{j=1}^n\sigma_j(\alpha) \right )^{[L:K(\alpha)]}</math>.

If ''L''/''K'' is [[Separable extension|separable]], then each root appears only once in the product (though the exponent, the [[Degree of a field extension|degree]] [''L'':''K''(''α'')], may still be greater than 1).