All quadratic non-residues in the Galois field.

An element \(x\) in \(\mathrm{GF}(p^m)\) is a quadratic non-residue if there does not exist a \(y\) such that \(y^2 = x\) in the field.

Examples¶

In fields with characteristic 2, no elements are quadratic non-residues.

In [1]: GF = galois.GF(2**4)

In [2]: GF.quadratic_non_residues
Out[2]: GF([], order=2^4)

In fields with characteristic greater than 2, exactly half of the nonzero elements are quadratic non-residues.

In [3]: GF = galois.GF(11)

In [4]: GF.quadratic_non_residues
Out[4]: GF([ 2,  6,  7,  8, 10], order=11)