User:Marty/UCSC Math 110 Fall 2008/Homework 9

The following is the last homework assignment. It does not need to be typed using LaTex (though you are free to use LaTex if you wish).

=Legendre symbols, 6 points=

Compute the following Legendre symbols (using quadratic reciprocity):
 * 1)  Compute $\left( \frac{3}{101} \right)$.
 * 2)  Compute $\left( \frac{23}{37} \right)$.
 * 3)  Compute $\left( \frac{50}{97} \right)$.

=Squares and prime powers, 4 points=

Suppose that $p$ is a prime number, $p \neq 2$, $a$ is an integer, and $\left( \frac{a}{p} \right) = 1$ (i.e., $a$ is a quadratic residue, mod $p$). Prove that $a$ is a quadratic residue, mod $p^2$. In other words, prove that there exists a residue $\bar b$, modulo $p^2$, such that $\bar b^2 = \bar a$, modulo $p^2$. For extra credit, prove that $a$ is a quadratic residue, mod $p^3$!

Hint:
 * Since $a$ is a quadratic residue mod $p$, you can find a residue $\bar c$ modulo $p$, such that $\bar c^2 = \bar a$, modulo $p$.
 * Try to modify $c$ (by adding a multiple of $p$), to obtain the desired residue $\bar b$.
 * You might try translating these statements into statements about divisibility of differences to help.