Quadratic field: Difference between revisions
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In [[mathematics]], a '''quadratic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] of its [[prime field]] of degree two. | In [[mathematics]], a '''quadratic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] of its [[prime field]] of degree two. | ||
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==Ring of integers== | ==Ring of integers== | ||
As above, we take ''d'' to be a square-free integer. The [[maximal order]] of ''F'' is | |||
:<math>O_F = \mathbf{Z}[\sqrt d] \,</math> | |||
unless <math>d \equiv 1 \pmod 4</math> in which case | |||
:<math>O_F = \mathbf{Z}\left[\frac{1+\sqrt d}{2}\right] .</math> | |||
===Discriminant=== | |||
The [[field discriminant]] of ''F'' is ''d'' if <math>d \equiv 1 \pmod 4</math> and otherwise 4''d''. | |||
===Unit group=== | ===Unit group=== | ||
If ''d'' is negative, then the only units in the ring of integers are [[root of unity|roots of unity]]. If ''d'' is positive, the unit group has rank one, with a [[fundamental unit]] of infinite order. | |||
===Class group=== | ===Class group=== | ||
==Splitting of primes== | ==Splitting of primes== | ||
The prime 2 is [[ramification|ramified]] if <math>d \equiv 2,3 \pmod 4</math>. If <math>d \equiv 1 \pmod 8</math> then 2 splits into two distinct prime ideals, and if <math>d \equiv 5 \pmod 8</math> then 2 is [[inertia|inert]]. | |||
An odd prime ''p'' [[ramification|ramifies]] iff ''p'' divides ''d''. Otherwise, ''p'' splits or is inert according as the [[Legendre symbol]] <math>\left(\frac{d}{p}\right)</math> is +1 or -1 respectively. | |||
==Galois group== | |||
The extension ''F''/'''Q''' is generated by the roots of <math>X^2 - d</math> and both roots lie in the field, which is thus a [[splitting field]] and so a [[Galois extension]]. The [[Galois group]] is cyclic of order two, with the non-trivial element being the [[field automorphism]] | |||
:<math>x + y\sqrt d \mapsto x - y\sqrt d .\,</math> | |||
If ''F'' is a complex quadratic field then this automorphism is induced by [[complex conjugation]]. | |||
==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 | pages=175-193,220-230,306-309 }} | |||
* {{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 | pages=59-62 }} | * {{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 | pages=59-62 }} | ||
* {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 | pages=34-36 }}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:01, 8 October 2024
In mathematics, a quadratic field is a field which is an extension of its prime field of degree two.
In the case when the prime field is finite, so is the quadratic field, and we refer to the article on finite fields. In this article we treat quadratic extensions of the field Q of rational numbers.
In characteristic zero, every quadratic equation is soluble by taking one square root, so a quadratic field is of the form for a non-zero non-square rational number d. Multiplying by a square integer, we may assume that d is in fact a square-free integer.
Ring of integers
As above, we take d to be a square-free integer. The maximal order of F is
![{\displaystyle O_{F}=\mathbf {Z} [{\sqrt {d}}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59270e30df17e18edae4b978241daba0fce3c93e)
unless in which case
![{\displaystyle O_{F}=\mathbf {Z} \left[{\frac {1+{\sqrt {d}}}{2}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab376e39f7a42a18ad01e5e988ed41f0f85df85)
Discriminant
The field discriminant of F is d if and otherwise 4d.
Unit group
If d is negative, then the only units in the ring of integers are roots of unity. If d is positive, the unit group has rank one, with a fundamental unit of infinite order.
Class group
Splitting of primes
The prime 2 is ramified if . If then 2 splits into two distinct prime ideals, and if then 2 is inert.
An odd prime p ramifies iff p divides d. Otherwise, p splits or is inert according as the Legendre symbol is +1 or -1 respectively.
Galois group
The extension F/Q is generated by the roots of and both roots lie in the field, which is thus a splitting field and so a Galois extension. The Galois group is cyclic of order two, with the non-trivial element being the field automorphism
If F is a complex quadratic field then this automorphism is induced by complex conjugation.
References
- A. Fröhlich; M.J. Taylor (1991). Algebraic number theory. Cambridge University Press, 175-193,220-230,306-309. ISBN 0-521-36664-X.
- I.N. Stewart; D.O. Tall (1979). Algebraic number theory. Chapman and Hall, 59-62. ISBN 0-412-13840-9.
- Pierre Samuel (1972). Algebraic number theory. Hermann/Kershaw, 34-36.