Quadratic field: Difference between revisions

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imported>Richard Pinch
(added ref Fröhlich+Taylor; splitting of primes)
imported>Richard Pinch
(References: added Samuel)
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==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 | year=1991 | isbn=0-521-36664-X | pages=175-193,220-230,306-309 }}
* {{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 }}

Revision as of 12:37, 7 December 2008

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

unless in which case

Discriminant

The field discriminant of F is d if and otherwise 4d.

Unit group

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.

References