Implement quadratic_nonresidue for more fields

[user202729]

Problem Description

Currently quadratic_nonresidue is implemented for some finite fields, but not others.

Having this method could simplify some part of the code

sage/src/sage/schemes/elliptic_curves/ell_field.py

Lines 164 to 170 in 1ca4a47

	# We could take a multiplicative generator but
	# that might be expensive to compute; otherwise
	# half the elements will do, and testing squares
	# is very fast.
	D = K.random_element()
	while D.is_square():
	D = K.random_element()

Somewhat related #39688

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

[Krishnadubey1008]

@user202729 Am I correctly getting your point that we only need to add the below method in ell_field.py

def quadratic_nonresidue(K):
    D = K.random_element()
    while D.is_square():
        D = K.random_element()
    return D

[vincentmacri]

Somewhat related #39688

Very related if we were to implement Tonelli's algorithm as (one of) the algorithm(s) to resolve #39688, since that algorithm needs a quadratic non-residue.

Am I correctly getting your point that we only need to add the below method in ell_field.py

I think it would rather be to implement it under ElementMethods in src/sage/categories/finite_fields.py.