Number field

Quadratic field

A quadratic field is a number field of degree 2, alg i.e. whence for some squarefree .

Proof

Let be a -basis for , where without loss of generality is an algebraic integer, whence for some . Let , so , and clearly is also a -basis for . Setting where and is squarefree, we have , so .

The ring of integers of a quadratic field are the Quadratic integers, whose structure is largely determined by mod . Any number which is an element of a quadratic field is a quadratic number.

Properties

1. By quadratic integers, is a monogenic field unless .

Classification

• Real quadratic field
• Imaginary quadratic field

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