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I have a question about Legendre symbol. Let $a$, $b$ be integers. Let $p$ be a prime not dividing $a$. Show that the Legendre symbol verifies: $$\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0.$$

I know that $\displaystyle\sum_{m=0}^{p-1} \left(\frac{m}{p}\right)=0$, but how do I connect this with the previous formula? Any help is appreciated.

Martin Sleziak
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kira
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    Show that as $m$ ranges from $0$ to $p-1$, $am$ ranges over all residue classes modulo $p$, and hence $am+b$ ranges over all residue classes modulo $p$. ā€“ Arturo Magidin Mar 10 '11 at 16:38
  • See also http://math.stackexchange.com/questions/1530467/sum-of-legendre-symbols-sum-n-1p-left-fracanbp-right-0 and http://math.stackexchange.com/questions/1666704/a-problem-with-the-legendre-jacobi-symbols-sum-n-1p-left-fracanbp ā€“ Martin Sleziak Feb 22 '16 at 14:12

1 Answers1

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To allow the question to be marked as answered, then:

Show that as $m$ ranges from $0$ to $pāˆ’1$, $am$ ranges over all residue classes modulo $p$, and hence $am+b$ ranges over all residue classes modulo $p$.

Arturo Magidin
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