Cyclotomic field: Difference between revisions
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In [[mathematics]], a '''cyclotomic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] generated by [[root of unity|roots of unity]]. If ζ denotes an ''n''-th root of unity, then the ''n''-th cyclotomic field ''F'' is the [[field extension]] <math>\mathbf{Q}(\zeta)</math>. | In [[mathematics]], a '''cyclotomic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] generated by [[root of unity|roots of unity]]. If ζ denotes an ''n''-th root of unity, then the ''n''-th cyclotomic field ''F'' is the [[field extension]] <math>\mathbf{Q}(\zeta)</math>. | ||
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==Splitting of primes== | ==Splitting of primes== | ||
A prime ''p'' [[ramification|ramifies]] iff ''p'' divides ''n''. Otherwise, the splitting of ''p'' depends on the common factor of ''p''-1 and ''n''. | A prime ''p'' [[ramification|ramifies]] iff ''p'' divides ''n''. Otherwise, the splitting of ''p'' depends on the factorisation of the polynomial <math>X^n-1</math> modulo ''p'', which in turn depends on the [[highest common factor]] of ''p''-1 and ''n''. | ||
==Galois group== | |||
The [[minimal polynomial]] for ζ is the ''n''-th [[cyclotomic polynomial]] <math>\Phi_n(X)</math>, which is a factor of <math>X^n-1</math>. Since the powers of ζ are the roots of the latter polynomial, ''F'' is a [[splitting field]] for <math>\Phi_n(X)</math> and hence a [[Galois extension]]. The [[Galois group]] is [[group isomorphism|isomorphic]] to the [[multiplicative group]], <math>(\mathbf{Z}/n\mathbf{Z})^*</math> via | |||
:<math>a \bmod n \leftrightarrow \sigma_a = (\zeta\mapsto\zeta^a) .\,</math> | |||
==References== | ==References== | ||
* {{cite book | author=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-36664-X }} | * {{cite book | author=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-36664-X }} | ||
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Cyclotomic Fields I and II | edition=Combined 2nd edition | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics]] | volume=121 | date=1990 | isbn=0-387-96671-4 }} | |||
* {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }} | |||
* {{cite book | author=I.N. Stewart | authorlink=Ian Stewart (mathematician) | coauthors=D.O. Tall | title=Algebraic number theory | publisher=Chapman and Hall | year=1979 | isbn=0-412-13840-9 }} | * {{cite book | author=I.N. Stewart | authorlink=Ian Stewart (mathematician) | coauthors=D.O. Tall | title=Algebraic number theory | publisher=Chapman and Hall | year=1979 | isbn=0-412-13840-9 }} | ||
* {{cite book | author= | * {{cite book | author=Lawrence C. Washington | authorlink=Lawrence C. Washington | title=Introduction to Cyclotomic Fields | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics]] | volume=83 | date=1982 | isbn=0-387-90622-3 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 3 August 2024
In mathematics, a cyclotomic field is a field which is an extension generated by roots of unity. If ζ denotes an n-th root of unity, then the n-th cyclotomic field F is the field extension .
Ring of integers
As above, we take ζ to denote an n-th root of unity. The maximal order of F is
![{\displaystyle O_{F}=\mathbf {Z} [\zeta ].\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f91377736d8bdf5904e6fd40e8ef2b3c17cce269)
Unit group
Class group
Splitting of primes
A prime p ramifies iff p divides n. Otherwise, the splitting of p depends on the factorisation of the polynomial modulo p, which in turn depends on the highest common factor of p-1 and n.
Galois group
The minimal polynomial for ζ is the n-th cyclotomic polynomial , which is a factor of . Since the powers of ζ are the roots of the latter polynomial, F is a splitting field for and hence a Galois extension. The Galois group is isomorphic to the multiplicative group, via
References
- A. Fröhlich; M.J. Taylor (1991). Algebraic number theory. Cambridge University Press. ISBN 0-521-36664-X.
- Serge Lang (1990). Cyclotomic Fields I and II, Combined 2nd edition. Springer-Verlag. ISBN 0-387-96671-4.
- Pierre Samuel (1972). Algebraic number theory. Hermann/Kershaw.
- I.N. Stewart; D.O. Tall (1979). Algebraic number theory. Chapman and Hall. ISBN 0-412-13840-9.
- Lawrence C. Washington (1982). Introduction to Cyclotomic Fields. Springer-Verlag. ISBN 0-387-90622-3.