A polynomial resultant is a single scalar modulo the coefficient-field prime. It vanishes when two input polynomials share a factor, and provides a practical test for square roots in finite-field extensions. section 12.0 section 12.1
For a polynomial A modulo an irreducible polynomial f over a field of prime p, compute res(f, A) and test res(f, A)^((p−1)/2) = 1. Equality identifies A as a quadratic residue and permits a square root; otherwise, the square root routine rejects it. section 12.1 section 12.4
Implementing the resultant follows a four-phase flow:
- Return zero immediately when either input polynomial is zero. section 12.2.3
- Compute each polynomial’s content, divide it out, initialize scaling factors, and swap inputs when the first degree is smaller. section 12.1
- Repeat pseudo-division while the second polynomial has positive degree, updating signs, leading coefficients, scaling values, and the remainder. section 12.2.3
Content normalization
For a = 4x⁴ + 24x³ + 12x² + 4x + 32, the content is 4. Dividing every coefficient by 4 produces the primitive polynomial used during resultant computation. section 12.1
