In which we prove quadratic reciprocity for the Jacobi symbol, and meet binary quadratic forms.
- Theorem 28 (Quadratic reciprocity for the Jacobi symbol): Let and be odd natural numbers. Then . We saw that we could prove this using just the definition of the Jacobi symbol and the law of quadratic reciprocity for the Legendre symbol (and a bit of book-keeping). So the law of quadratic reciprocity for the Legendre symbol is the really interesting result; the version for the Jacobi symbol was a pretty straightforward consequence.
- Definition of binary quadratic forms.
- Definition of unimodular substitutions, and equivalence of forms.
- Exercise: Check that equivalence of forms is an equivalence relation.
- Definition of the discriminant of a form.
- Lemma 29: Equivalent forms have the same discriminant. We saw that we could prove this by direct calculation, or by using the matrix interpretation. Is the converse true?
Davenport (The Higher Arithmetic) and Baker (A concise introduction to the theory of numbers) both have chapters on binary quadratic forms that cover the material in this section of the course.
Preparation for Lecture 10
You are now well placed to tackle some questions about binary quadratic forms that we shall address in the next lecture or two.
- If two binary quadratic forms have the same discriminant, must they be equivalent?
- Which numbers can be the discriminant of a binary quadratic form?
- We have an equivalence relation on the collection of binary quadratic forms, and so we have equivalence classes of forms. It would be useful to be able to pick exactly one (hopefully quite nice) representative of each equivalence class. How might we do that? Can we somehow ‘reduce’ each form?
- Our big aim for this section is to be able to decide which numbers are represented by which forms. You could start (continue?) to try to make some conjectures about this, and of course to try to prove those conjectures.