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.

• Lemma 27: Let $n$ be an odd natural number.  Then $\genfrac{(}{)}{}{}{-1}{n} = (-1)^{\frac{n-1}{2}}$ and $\genfrac{(}{)}{}{}{2}{n} = (-1)^{\frac{n^2-1}{8}}$.  We proved this using the definition of the Jacobi symbol and the corresponding results for the Legendre symbol.
• Theorem 28 (Law of quadratic reciprocity for the Jacobi symbol): Let $m$ and $n$ be odd natural numbers.  Then $\genfrac{(}{)}{}{}{m}{n} = (-1)^{\frac{m-1}{2} \frac{n-1}{2}} \genfrac{(}{)}{}{}{n}{m}$.  We proved this using the definition of the Jacobi symbol and the corresponding result for 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 $n$?

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?)
• 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 $SL_2(\mathbb{Z})$)?