A report on Binary quadratic form

Quadratic homogeneous polynomial in two variables

- Binary quadratic form

6 related topics with Alpha

Overall

Bhargava cube with the integers a, b, c, d, e, f, g, h at the corners

Bhargava cube

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Configuration consisting of eight integers placed at the eight corners of a cube.

Configuration consisting of eight integers placed at the eight corners of a cube.

Bhargava cube with the integers a, b, c, d, e, f, g, h at the corners
An example of Bhargava cube

To each pair of opposite faces of a Bhargava cube one can associate an integer binary quadratic form thus getting three binary quadratic forms corresponding to the three pairs of opposite faces of the Bhargava cube.

Genus of a quadratic form

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Classification of quadratic forms and lattices over the ring of integers.

Classification of quadratic forms and lattices over the ring of integers.

For binary quadratic forms there is a group structure on the set C of equivalence classes of forms with given discriminant.

Fermat's theorem on sums of two squares

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Odd prime p can be expressed as:

Odd prime p can be expressed as:

An (integral binary) quadratic form is an expression of the form, as required.

Manjul Bhargava in 2014

Manjul Bhargava

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Canadian-American mathematician.

Canadian-American mathematician.

Manjul Bhargava in 2014
Manjul Bhargava in 2014
Four Fields medallists left to right (Artur Avila, Martin Hairer (at back), Maryam Mirzakhani, with Maryam's daughter Anahita) and Bhargava at the ICM 2014 in Seoul

His PhD thesis generalized Gauss's classical law for composition of binary quadratic forms to many other situations.

Infrastructure (number theory)

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Infrastructure is a group-like structure appearing in global fields.

Infrastructure is a group-like structure appearing in global fields.

D. Shanks observed the infrastructure in real quadratic number fields when he was looking at cycles of reduced binary quadratic forms.

Ideal class group

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Algebraic number field

Algebraic number field

For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic forms of discriminant equal to the discriminant of Q(√d).