Quadratic field

Quadratic integers

The quadratic integers within a quadratic field where is a squarefree integer are ring

Proof

Let Clearly an element of of degree 1 is an algebraic integer iff it is an integer. Let be a degree 2 element. Then the minimal polynomial of is

By ^P1, we have iff , which is precisely the case when . It follows , so since is squarefree . Letting , , we have iff and . Since is squarefree it follows , so we need only consider the cases

1. If then which holds iff ;
2. If then which holds iff ;
3. If then which holds iff .

It follows that the general expression for an algebraic integer is

• if
• if

where , whence the above.

In general, a quadratic integer is the solution to some monic quadratic with integer coëfficients.

Properties

Let be a (proper) quadratic integer with minimal polynomial

1. The discriminant is .
2. It follows that

Proof of 1

See Discriminant of an algebraic integer.

Prime ideals

Let be an odd prime and be the corresponding Legendre symbol.

1. If then is unramified at , where .
2. If then is inert at .¹

Proof

First suppose for . Then

and on the other hand contains both and . Thus by Bézout’s lemma we have , so .

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Footnotes

1. 2022. Algebraic number theory course notes, ¶2.12, pp. 38–39. ↩