Editing Field norm

{{Short description|Concept in field theory mathematics}}
In [[mathematics]], the '''(field) norm''' is a particular mapping defined in [[field theory (mathematics)|field theory]], which maps elements of a larger field into a [[field extension|subfield]].

==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).

==Examples==

=== Quadratic field extensions ===
One of the basic examples of norms comes from [[quadratic field]] extensions <math>\Q(\sqrt{a})/\Q</math> where <math>a</math> is a square-free integer.

Then, the multiplication map by <math>\sqrt{a}</math> on an element <math>x + y \cdot \sqrt{a}</math> is
:<math>\sqrt{a}\cdot (x + y\cdot\sqrt{a}) = y \cdot a +  x \cdot \sqrt{a}.</math>
The element <math>x + y \cdot \sqrt{a}</math> can be represented by the vector
:<math>\begin{bmatrix}x \\ y\end{bmatrix},</math>
since there is a direct sum decomposition <math>\Q(\sqrt{a}) = \Q\oplus \Q\cdot\sqrt{a}</math> as a <math>\Q</math>-vector space.

The [[matrix (mathematics)|matrix]] of <math>m_\sqrt{a}</math> is then
:<math>m_{\sqrt{a}} = \begin{bmatrix}
0 & a \\
1 & 0
\end{bmatrix}</math>
and the norm is <math>N_{\Q(\sqrt{a})/\Q}(\sqrt{a}) = -a</math>, since it is the determinant of this matrix.

==== Norm of Q(√2) ====

Consider the [[algebraic number field|number field]] <math>K=\Q(\sqrt{2})</math>.

The Galois group of <math>K</math> over <math>\Q</math> has order <math>d = 2</math> and is generated by the element which sends <math>\sqrt{2}</math> to <math>-\sqrt{2}</math>. So the norm of <math>1+\sqrt{2}</math> is:

:<math>(1+\sqrt{2})(1-\sqrt{2}) = -1.</math>

The field norm can also be obtained without the Galois group.

Fix a <math>\Q</math>-basis of <math>\Q(\sqrt{2})</math>, say:
:<math>\{1,\sqrt{2}\}</math>.

Then multiplication by the number <math>1+\sqrt{2}</math> sends
:1 to <math>1+\sqrt{2}</math> and
:<math>\sqrt{2}</math> to <math>2+\sqrt{2}</math>.

So the determinant of "multiplying by <math>1+\sqrt{2}</math>" is the determinant of the matrix which sends the vector
:<math>\begin{bmatrix}1 \\ 0\end{bmatrix}</math> (corresponding to the first basis element, i.e., 1) to <math>\begin{bmatrix}1 \\ 1\end{bmatrix}</math>,
:<math>\begin{bmatrix}0 \\ 1\end{bmatrix}</math> (corresponding to the second basis element, i.e., <math>\sqrt{2}</math>) to <math>\begin{bmatrix}2 \\ 1\end{bmatrix}</math>,
viz.:

:<math>\begin{bmatrix}1 & 2 \\1 & 1 \end{bmatrix}.</math>

The determinant of this matrix is −1.

=== ''p''-th root field extensions ===
Another easy class of examples comes from field extensions of the form <math>\mathbb{Q}(\sqrt[p]{a})/\mathbb{Q}</math> where the prime factorization of <math>a \in \mathbb{Q} </math> contains no <math>p</math>-th powers, for <math>p</math> a fixed odd prime.

The multiplication map by <math>\sqrt[p]{a}</math> of an element is<blockquote><math>\begin{align}
m_{\sqrt[p]{a}}(x) &= \sqrt[p]{a} \cdot (a_0 + a_1\sqrt[p]{a} + a_2\sqrt[p]{a^2} + \cdots + a_{p-1}\sqrt[p]{a^{p-1}} )\\
&= a_0\sqrt[p]{a} + a_1\sqrt[p]{a^2} + a_2\sqrt[p]{a^3} + \cdots + a_{p-1}a 
\end{align}</math></blockquote>giving the matrix<blockquote><math>\begin{bmatrix}
0 & 0 & \cdots & 0 & a \\
1 & 0 & \cdots & 0 & 0 \\
0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots  & \vdots & \vdots \\
0 & 0 & \cdots & 1 & 0
\end{bmatrix}</math></blockquote>The determinant gives the norm
:<math>N_{\mathbb{Q}(\sqrt[p]{a})/\mathbb{Q}}(\sqrt[p]{a}) = (-1)^{p-1} a = a.</math>

=== Complex numbers over the reals ===
The field norm from the [[complex number]]s to the [[real number]]s sends

: {{nowrap|''x'' + ''iy''}}

to

: {{nowrap|''x''<sup>2</sup> + ''y''<sup>2</sup>}},

because the Galois group of <math>\Complex</math> over <math>\R</math> has two elements,
* the identity element and
* complex conjugation,
and taking the product yields {{nowrap|1=(''x'' + ''iy'')(''x'' − ''iy'') = ''x''<sup>2</sup> + ''y''<sup>2</sup>}}.

=== Finite fields ===
Let ''L'' = GF(''q''<sup>''n''</sup>) be a finite extension of a [[finite field]] ''K'' = GF(''q'').

Since ''L''/''K'' is a Galois extension, if ''α'' is in ''L'', then the norm of ''α'' is the product of all the Galois conjugates of ''α'', i.e.<ref name="LN57">{{harvnb|Lidl|Niederreiter|1997|loc=p. 57}}</ref>

: <math> \operatorname{N}_{L/K}(\alpha)=\alpha \cdot \alpha^q \cdot \alpha^{q^2} \cdots \alpha^{q^{n-1}} = \alpha^{(q^n - 1)/(q-1)}. </math>

In this setting we have the additional properties,<ref>{{harvnb|Mullen|Panario|2013|loc=p. 21}}</ref>

*<math>\forall \alpha \in L, \quad \operatorname{N}_{L/K}(\alpha^q) = \operatorname{N}_{L/K}(\alpha) </math>
*<math>\forall a \in K, \quad \operatorname{N}_{L/K}(a) = a^n.</math>

==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.

==Further properties==
The norm of an [[algebraic integer]] is again an integer, because it is equal (up to sign) to the constant term of the characteristic polynomial.

In [[algebraic number theory]] one defines also norms for [[ideal (ring theory)|ideal]]s. This is done in such a way that if ''I'' is a nonzero ideal of ''O''<sub>''K''</sub>, the ring of integers of the number field ''K'', '''N'''(''I'') is the number of residue classes in <math>O_K / I</math>&nbsp;– i.e. the cardinality of this [[finite ring]]. Hence this [[ideal norm]] is always a positive integer.

When ''I'' is a [[principal ideal]] ''αO<sub>K</sub>'' then '''N'''(''I'') is equal to the [[absolute value]] of the norm to ''Q'' of ''α'', for ''α'' an [[algebraic integer]].

==See also==
* [[Field trace]]
* [[Ideal norm]]
* [[Norm form]]

==Notes==
{{reflist|3}}

==References==
* {{citation | first1=Rudolf | last1=Lidl | first2=Harald | last2=Niederreiter | author2-link=Harald Niederreiter | title=Finite Fields | series=Encyclopedia of Mathematics and its Applications | volume=20 | year=1997 | orig-year=1983 | edition=Second | publisher=[[Cambridge University Press]] | isbn=0-521-39231-4 | zbl=0866.11069 | url-access=registration | url=https://archive.org/details/finitefields0000lidl_a8r3 }}
* {{citation|first1=Gary L.|last1=Mullen|first2=Daniel|last2=Panario|title=Handbook of Finite Fields|year=2013|publisher=CRC Press|isbn=978-1-4398-7378-6}}
* {{citation | last=Roman | first=Steven | title=Field theory | edition=Second | year=2006 | publisher=Springer | series=[[Graduate Texts in Mathematics]] | volume=158 | at=Chapter 8 | isbn=978-0-387-27677-9 | zbl=1172.12001 }}
* {{citation|first=Joseph J.|last=Rotman|title=Advanced Modern Algebra|year=2002|publisher=Prentice Hall|isbn=978-0-13-087868-7}}

[[Category:Algebraic number theory]]