Number Theory: Lecture 8

In which we meet the Jacobi symbol, and see which of our results about the Legendre symbol transfer across to the Jacobi symbol.

  • Definition of the Jacobi symbol.
  • Really important remark: the Jacobi symbol \genfrac{(}{)}{}{}{a}{n} does not record whether a is a quadratic residue modulo n.
  • Lemma 26 (Multiplicativity of the Jacobi symbol): The Jacobi symbol is totally multiplicative in two senses.
  1. If n is an odd natural number and if a and b are integers, then \genfrac{(}{)}{}{}{ab}{n} = \genfrac{(}{)}{}{}{a}{n} \genfrac{(}{)}{}{}{b}{n}.
  2. If m and n are odd natural numbers and if a is an integer, then \genfrac{(}{)}{}{}{a}{mn} = \genfrac{(}{)}{}{}{a}{m} \genfrac{(}{)}{}{}{a}{n}.

Both proofs followed very quickly from the definition of the Jacobi symbol and the total multiplicativity of the Legendre symbol.

Understanding today’s lecture

Pick some Jacobi symbols to compute explicitly.  Can you find examples so that you use all of the results we proved about the Jacobi symbol?

Can you find another example (in addition to the one we saw in lectures) of a Jacobi symbol where \genfrac{(}{)}{}{}{a}{n} = 1 but a is not a quadratic residue modulo n?

Further reading

The same selection that I suggested last time.

Preparation for Lecture 9

Next time, we’ll start a section of the course in which we’ll study binary quadratic forms.  These are objects of the form ax^2 + bxy + cy^2, where the coefficients a, b and c are integers (and we think of x and y as integer variables).  We are going to be interested in questions to do with which numbers can be represented by such forms (that is, as x and y range over the integers, which values do we get from ax^2 + bxy + cy^2).  Here are some questions that you could usefully try yourself before the lecture.

  • Which numbers are represented by the form 4x^2 + 12xy + 9y^2?
  • Which numbers are represented by the form x^2 + y^2?  (That is, which numbers can be written as a sum of two squares?)
  • What about other forms?  Try your own example(s).
  • When do two forms represent the same set of numbers?  For example, can you find any forms that give the same set of numbers as the form 4x^2 + 12xy + 9y^2, or the same set of the numbers as the form x^2 + y^2?
  • What is the link between binary quadratic forms ax^2 + bxy + cy^2 and 2 \times 2 integer matrices with determinant 1 (that is, elements of SL_2(\mathbb{Z}))?
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