Editing Discriminant (section)

===Fundamental discriminants===

A specific type of discriminant useful in the study of quadratic fields is the fundamental discriminant. It arises in the theory of integral [[Binary quadratic form|binary quadratic forms]], which are expressions of the form:<math display="block">Q(x, y) = ax^2 + bxy + cy^2</math>


where <math display="inline">a</math>, <math display="inline">b</math>, and <math display="inline">c</math> are integers. The discriminant of <math display="inline">Q(x, y)</math> is given by:<math display="block">D = b^2 - 4ac</math>Not every integer can arise as a discriminant of an integral binary quadratic form. An integer <math display="inline">D</math> is a fundamental discriminant if and only if it meets one of the following criteria:

* Case 1: <math display="inline">D</math> is congruent to 1 modulo 4 (<math display="inline">D \equiv 1 \pmod{4}</math>) and is square-free, meaning it is not divisible by the square of any prime number.
* Case 2: <math display="inline">D</math> is equal to four times an integer <math display="inline">m</math> (<math display="inline">D = 4m</math>) where <math display="inline">m</math> is congruent to 2 or 3 modulo 4 (<math display="inline">m \equiv 2, 3 \pmod{4}</math>) and is square-free.

These conditions ensure that every fundamental discriminant corresponds uniquely to a specific type of quadratic form.

The first eleven positive fundamental discriminants are:

: [[1 (number)|1]], [[5 (number)|5]], [[8 (number)|8]], [[12 (number)|12]], [[13 (number)|13]], [[17 (number)|17]], [[21 (number)|21]], [[24 (number)|24]], [[28 (number)|28]], [[29 (number)|29]], [[33 (number)|33]] (sequence A003658 in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

The first eleven negative fundamental discriminants are:

: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

==== Quadratic number fields ====
A quadratic field is a field extension of the rational numbers <math display="inline">\mathbb{Q}</math> that has degree 2. The discriminant of a quadratic field plays a role analogous to the discriminant of a quadratic form.

There exists a fundamental connection: an integer <math display="inline">D_0</math> is a fundamental discriminant if and only if:

* <math display="inline">D_0 = 1</math>, or
* <math display="inline">D_0</math> is the discriminant of a quadratic field.

For each fundamental discriminant <math display="inline">D_0 \neq 1</math>, there exists a unique (up to isomorphism) quadratic field with <math display="inline">D_0</math> as its discriminant. This connects the theory of quadratic forms and the study of quadratic fields.

==== Prime factorization ====
Fundamental discriminants can also be characterized by their prime factorization. Consider the set <math display="inline">S</math> consisting of <math>-8, 8, -4,</math> the prime numbers congruent to 1 modulo 4, and the [[additive inverse]]s of the prime numbers congruent to 3 modulo 4:<math display="block">S = \{-8, -4, 8, -3, 5, -7, -11, 13, 17, -19, ... \}</math>An integer <math display="inline">D \neq 1</math> is a fundamental discriminant if and only if it is a product of elements of <math>S</math> that are pairwise [[coprime]].{{cn|date=August 2024}}