Properties of Legendre Symbol

Number Theory · Axiom Academy

LESSON Properties of Legendre Symbol Exploring multiplicativity and special values that make quadratic reciprocity possible The Legendre symbol is multiplicative in the numerator : the symbol of a product equals the product of the symbols. This means if we know whether a and b are quadratic residues, we can immediately determine whether their product ab is a quadratic residue. 2. Dependence on Residue Class The Legendre symbol depends only on the residue class of a modulo p . If two numbers are congruent modulo p , they have the same Legendre symbol. This property allows us to reduce any integer to its representative in the range 1 to p -1 before computing the Legendre symbol. 3. First Supplement: Legendre Symbol of -1 The first supplement to quadratic reciprocity gives us an explicit formula for when -1 is a quadratic residue modulo a prime p . This remarkable result shows that -1 is a square modulo p if and only if p is congruent to 1 modulo 4. p = 5: Since 5 ≡ 1 (mod 4), we have (-1/5) = +1. Indeed, 2² = 4 ≡ -1 (mod 5). p = 7: Since 7 ≡ 3 (mod 4), we have (-1/7) = -1. So -1 is not a square mod 7. 4. Second Supplement: Legendre Symbol of 2 The second supplement to quadratic reciprocity tells us when 2 is a quadratic residue modulo an odd prime p . This shows that 2 is a square modulo p if and only if p is congruent to ±1 modulo 8. p = 7: Since 7 ≡ -1 ≡ 7 (mod 8), we have (2/7) = +1. Indeed, 3² = 9 ≡ 2 (mod 7).

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