Quadratic field: Difference between revisions
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In characteristic zero, every [[quadratic equation]] is soluble by taking one square root, so a quadratic field is of the form <math>\mathbf{Q}(\sqrt d)</math> 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]]. | In characteristic zero, every [[quadratic equation]] is soluble by taking one square root, so a quadratic field is of the form <math>\mathbf{Q}(\sqrt d)</math> 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== | |||
===Unit group=== | |||
===Class group=== | |||
==Splitting of primes== | |||
==References== | |||
* {{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 }} | |||
Revision as of 04:47, 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
Unit group
Class group
Splitting of primes
References
- I.N. Stewart; D.O. Tall (1979). Algebraic number theory. Chapman and Hall, 59-62. ISBN 0-412-13840-9.