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| In [[mathematics]], the '''(field) norm''' is a mapping defined in [[field theory (mathematics)|field theory]], to map elements of a larger field into a smaller one.
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| ==Formal definitions==
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| '''1.''' Let ''K'' be a field and ''L'' a finite [[field extension|extension]] (and hence an [[algebraic extension]]) of ''K''. Multiplication by α, an element of ''L'', is a ''K''-[[linear transformation]]
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| :<math>m_\alpha:L\to L\;.</math>
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| That is, ''L'' is viewed as a [[vector space]] over ''K'', and ''m''<sub>α</sub> is a 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. Properties of the determinant imply that the norm belongs to ''K'' and
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| :'''N'''<sub>''L/K''</sub>(αβ) = '''N'''<sub>''L/K''</sub>(α)'''N'''<sub>''L/K''</sub>(β)
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| so that the norm, when considered on non-zero elements, is a [[group homomorphism]] from the multiplicative group of ''L'' to that of ''K''.
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| '''2.''' If ''L/K'' is a [[Galois extension]], the norm '''N'''<sub>''L/K''</sub> of an element α of ''L'' is the product of all the [[Conjugate element (field theory)|conjugates]]
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| :''g''(α) | |
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| of α, for ''g'' in the [[Galois group]] ''G'' of ''L''/''K''.
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| ==Example==
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| The field norm from the [[complex number]]s to the [[real numbers]] sends
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| : ''x'' + ''iy''
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| to
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| : ''x<sup>2</sup>'' + ''y<sup>2</sup>''. This corresponds to the Galois group having two elements, the identity element which sends ''x'' + ''iy'' to itself, and complex conjugation, which sends ''x'' + ''iy'' to ''x'' − ''iy''.
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| Notice that in this case the norm is the square of the "usual" norm, in the sense that we would normally consider the above number ''x''+''iy'' as having norm equal to the Euclidean distance from the origin in the complex plane, which is <math>\sqrt{(x^2+y^2)}</math>. In some sense therefore we need to speak of this field norm as an `arithmetic norm' as opposed to the usual Euclidean (distance) norm.
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| This is particularly important in algebraic number theory where we may have a number field ''K'' (a finite extension of the rationals <math>\Q</math> which for simplicity we assume to be Galois), which is naturally itself a vector space of finite dimension ''d'' say over <math>\Q</math>. The field norm from ''K'' down to <math>\Q</math> of any element will be as defined above, namely the product of all of its ''d'' Galois conjugates. However we may also embed ''K'' in a standard way into the real or complex numbers <math>\mathbb{R}</math> or <math>\mathbb{C}</math> (according to whether ''K'' contains any imaginary numbers), and we then consider the [[Norm (mathematics)]] or [[absolute value]] of the element within the larger field in the usual way. This in general will be very different from the field norm. We give an example below.
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| Finally, if (as is common in algebraic number theory) we consider ''K'' as embedded inside a real vector space <math>K \otimes_\Q \mathbb{R} \cong \mathbb{R}^d </math> then both of these norms will in general be very different from the Euclidean norm inside <math>K \otimes_\Q \mathbb{R} </math> (which is basis-dependent - see below).
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| As an example of this, consider the number field <math>K=\Q(\sqrt{2})</math> obtained from <math>\Q</math> by adjoining <math>\sqrt{2}</math>, which is the splitting field for the polynomial <math>X^2-2</math>. The Galois group of this extension is of order ''d=2'', generated by the element which sends <math>\sqrt{2}</math> to <math>-\sqrt{2}</math>, and so the norm of the number <math>1+\sqrt{2}</math> is:
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| : <math>(1+\sqrt{2})*(1-\sqrt{2}) = -1.</math>
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| We may also see this using the terminology of "formal definition 1" above if we fix a basis for the extension <math>\Q(\sqrt{2}):\Q</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> while <math>\sqrt{2}</math> is mapped to <math>2+\sqrt{2}</math>. Hence the determinant of the mapping is that of the matrix which sends the vector <math>(1,0)^T</math> (corresponding to the first basis element, ie 1) to <math>(1,1)^T</math> and the vector <math>(0,1)^T</math> (which represents the second basis element <math>\sqrt{2}</math>) to <math>(2,1)^T</math>, viz.:
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| :<math>\begin{bmatrix}1 & 2 \\1 & 1 \end{bmatrix}. </math>
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| The determinant of this matrix is again -1, showing that the two "formal" definitions of norm agree. However, if we view ''K'' as a subfield of the complex numbers <math>\mathbb{C}</math> in the normal way, then the norm (ie the [[absolute value]]) is of course <math>1+\sqrt{2}</math>.
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| Finally if we embed ''K'' inside <math>K \otimes_\Q \mathbb{R} \cong \mathbb{R}^2 </math> then taking <math>\{1,\sqrt{2}\}</math> as a basis for ''K'' over <math>\Q</math> we see that <math>1+\sqrt{2}</math> is represented by the vector <math>(1,1)</math> and so its Euclidean norm inside <math> \mathbb{R}^2 </math> is <math>\sqrt{2}</math>.
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| ==Further properties==
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| 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.
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| 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 an ideal of ''O''<sub>''K''</sub>, the [[ring of integers]] of the [[number field]] ''K'', '''N'''(''I'') is the number of residue classes in ''O''<sub>''K''</sub>/''I'' – i.e. the cardinality of this [[finite ring]]. Hence this '''norm of an ideal''' is always a positive integer. When ''I'' is a principal ideal α''O''<sub>''K''</sub> then '''N'''(''I'') is equal to the [[absolute value]] of the norm to ''Q'' of α, for α an algebraic integer.
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| ==See also== | |
| * [[Field trace]]
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| * [[Ideal norm]]
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| *[[Norm field]]
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| ==References==
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| *{{Citation
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| | last=Roman
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| | first=Steven
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| | title=Field theory
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| | edition=Second
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| | year=2006
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| | publisher=Springer
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| | series=Graduate Texts in Mathematics
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| | volume=158
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| | at=Chapter 8
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| | isbn=978-0-387-27677-9
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| }}
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| {{DEFAULTSORT:Field Norm}}
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| [[Category:Field theory]]
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| [[Category:Algebraic number theory]]
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