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