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Number Theory

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Quadratic Reciprocity

NUMTH-BGJWZS

Suppose that $p$, $q$ are odd primes, and $x^2 \equiv p \ \mathrm{mod} \ q$ has a solution.

Which of the following statements are always correct? Select ALL that apply.

A

$x^2 \equiv q \ \mathrm{mod} \ p$ has a solution.

B

$p \equiv 1 \ \mathrm{mod} \ 4$.

C

$q \equiv 1 \ \mathrm{mod} \ 4$.

D

If $p\equiv 1 \ \mathrm{mod} \ 4$, then $x^2 \equiv q \ \mathrm{mod} \ p$ has a solution.

E

If $q\equiv 1 \ \mathrm{mod} \ 4$, then $x^2 \equiv q \ \mathrm{mod} \ p$ has a solution.