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| In [[algebraic number theory]], the '''narrow class group''' of a [[number field]] ''K'' is a refinement of the [[class group]] of ''K'' that takes into account some information about embeddings of ''K'' into the field of [[real number]]s.
| | Greetings! I am Myrtle Shroyer. His spouse doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been performing it for quite a while. Years ago we moved to North Dakota and I adore every working day residing here. Hiring is her working day job now and she will not change it whenever quickly.<br><br>Feel free to visit my website - [http://btcsoc.com/index.php?do=/profile-4621/info/ std testing at home] |
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| == Formal definition ==
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| Suppose that ''K'' is a [[finite extension]] of '''Q'''. Recall that the ordinary class group of ''K'' is defined to be
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| :<math>C_K = I_K / P_K,\,\!</math>
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| where ''I''<sub>''K''</sub> is the group of [[fractional ideal]]s of ''K'', and ''P''<sub>''K''</sub> is the group of principal fractional ideals of ''K'', that is, ideals of the form ''aO''<sub>''K''</sub> where ''a'' is a [[Unit (ring theory)|unit]] of ''K''.
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| The '''narrow class group''' is defined to be the quotient
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| :<math>C_K^+ = I_K / P_K^+,</math>
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| where now ''P''<sub>''K''</sub><sup>+</sup> is the group of '''totally positive principal fractional ideals''' of ''K''; that is, ideals of the form ''aO''<sub>''K''</sub> where ''a'' is a unit of ''K'' such that σ(''a'') is ''positive'' for every embedding
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| :<math>\sigma : K \to \mathbf R.</math>
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| == Uses ==
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| The narrow class group features prominently in the theory of representing of integers by [[quadratic form]]s. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25).
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| :'''Theorem'''. Suppose that
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| ::<math>K = \mathbf Q(\sqrt d),</math>
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| :where ''d'' is a [[square-free integer]], and that the narrow class group of ''K'' is trivial. Suppose that
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| ::<math>\{ \omega_1, \omega_2 \}\,\!</math>
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| :is a basis for the ring of integers of ''K''. Define a quadratic form
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| :: <math>q_K(x,y) = N_{K/\mathbf Q}(\omega_1 x + \omega_2 y)</math>,
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| :where ''N''<sub>''K''/'''Q'''</sub> is the [[field norm|norm]]. Then a [[prime number]] ''p'' is of the form
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| :: <math>p = q_K(x,y)\,\!</math>
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| :for some integers ''x'' and ''y'' '''if and only if''' either
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| :: <math>p \mid d_K\,\!,</math>
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| :or
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| :: <math>p = 2 \quad \mbox{and} \quad d_K \equiv 1 \pmod 8,</math>
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| :or
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| ::<math> p > 2 \quad \mbox{and} \quad \left(\frac {d_K} p\right) = 1,</math>
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| :where ''d''<sub>''K''</sub> is the [[Discriminant#Discriminant of an algebraic number field|discriminant]] of ''K'', and
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| ::<math>\left(\frac ab\right)</math>
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| :indicates the [[Legendre symbol]].
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| === Examples ===
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| For example, one can prove that the quadratic fields '''Q'''(√−1), '''Q'''(√2), '''Q'''(√−3) all have trivial narrow class group. Then, by choosing appropriate bases for the integers of each of these fields, the above theorem implies the following:
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| * A prime ''p'' is of the form ''p'' = ''x''<sup>2</sub> + ''y''<sup>2</sub> for integers ''x'' and ''y'' if and only if
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| ::<math>p = 2 \quad \mbox{or} \quad p \equiv 1 \pmod 4.</math>
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| : (This is known as [[Fermat's theorem on sums of two squares]].)
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| * A prime ''p'' is of the form ''p'' = ''x''<sup>2</sub> − 2''y''<sup>2</sub> for integers ''x'' and ''y'' if and only if
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| ::<math>p = 2 \quad \mbox{or} \quad p \equiv 1, 7 \pmod 8.</math>
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| * A prime ''p'' is of the form ''p'' = ''x''<sup>2</sub> − ''xy'' + ''y''<sup>2</sub> for integers ''x'' and ''y'' if and only if
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| ::<math>p = 3 \quad \mbox{or} \quad p \equiv 1 \pmod 3.</math> (cf. [[Eisenstein prime]])
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| == See also ==
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| * [[Class group]]
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| * [[Quadratic form]]
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| == References ==
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| * A. Fröhlich and M. J. Taylor, ''Algebraic Number Theory'' (p.180), Cambridge University Press, 1991.
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| [[Category:Algebraic number theory]]
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Greetings! I am Myrtle Shroyer. His spouse doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been performing it for quite a while. Years ago we moved to North Dakota and I adore every working day residing here. Hiring is her working day job now and she will not change it whenever quickly.
Feel free to visit my website - std testing at home