General number fields and quadratic forms in n variables.

Example of binary quadratic forms. The foundational Disquisitiones Arithmeticae Gau86 of Gauss has also been used to study reduction of indefinite forms along with Coh93. B1 BinaryQF 1 2 3 sage. That is there is exactly one reduced binary quadratic forms of discriminant 4 namely x2 y2 and hence just one equivalence class of binary quadratic forms of discriminant.

We say that fis primitive if gcdabc 1. The references are Hec10 Ch. For example as the integer 5 is represented by 3a22ab5b2 namely with ab01 it is also represented by 3a28ab10b2 with ab-11.

Binary Quadratic Forms The most famous result in elementary number theory involving binary quadratic forms is Fermats Two-Squares Theorem. Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus. We are interested in what numbers can be represented in a given quadratic form.

If it is congruent to 1 modulo 4. Discriminant -8 sage. B BinaryQF 1 2 3 sage.

B1 b True sage. For example for d 4 a 1 so that a 1 and a b a with b2 4 divisible by 4 so that b 0 and hence c 1. For binary quadratic forms there is a group structure on the set C equivalence classes of forms with given discriminant.

We have that a 4 b -24 and c 39. 1 2 for repre-sentation problems. This theory is elementary but quite powerful and to develop it we rst must de ne some.