There are algebraic formulas, e.g.
a = k*(p^2  q^2), b = k*(2*p*q), c = k*(p^2 + q^2)
for which a^2 + b^2 = c^2 is a polynomial identity.
So k, p, and q can be rational integers, Gaussian integers, Eisenstein integers, arbitrary complex numbers, or just variables  it's all good.
There is a book entitled Mathematics, Its Magic And Mastery by Aaron Bakst. One of its chapters is entitled Algebra, Boss of Arithmetic.
