Primitive polynomials with prescribed second coefficient

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed. This has recently been proved whenever n ≥ 9. It is also known to be true when n ≤ 3. We show that there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of x^n-2) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x ⁴ + a₁x³ + x² + a₃x + 1 over the binary field. For ≥ 6 we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.