In 2005, the only surviving portrait of this mathematician was discovered to actually be of an obscure politician who shares his last name. For 10 points each:
[10e] Name this French mathematician whose namesake symbol is equivalent to the Jacobi symbol if p is an odd prime. His namesake transform converts a function of one set of variables to a function of a conjugate set of variables, producing the Hamiltonian from the Lagrangian.
ANSWER: Adrien-Marie Legendre [accept Legendre symbol or Legendre transform]
[10m] The Legendre symbol returns positive 1 if a is one of these values modulo p, meaning that there exists a perfect square x such that x is congruent to a mod p.
ANSWER: quadratic residue [prompt on residues]
[10h] One can determine if an integer is a quadratic residue using Euler’s criterion, which states that the Legendre symbol of a over p is congruent to this function of a and p, mod p.
ANSWER: a raised to the power p minus 1 over 2 [accept answers that use one-half p minus one-half]
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