The fact that a polynomial q(z) might not be linear is irrelevant to whether the set of all polynomials 𝔽[z] forms a vector space over 𝔽.

Here's why 𝔽[z] forms a vector space over 𝔽, regardless of individual polynomial degrees:

Vector Space Requirements

A vector space is a set of elements (vectors) along with two operations: addition of vectors and multiplication of vectors by scalars (from a field). This set and its operations must satisfy a list of axioms. The key axioms include:

1. Closure under addition: The sum of any two polynomials is another polynomial.
2. Associativity of addition: (p + q) + r = p + (q + r) for all polynomials p, q, r.
3. Commutativity of addition: p + q = q + p for all polynomials p, q.
4. Existence of a zero vector: The zero polynomial (all coefficients 0) acts as an additive identity.
5. Existence of additive inverses: For every polynomial p, there exists a polynomial -p such that p + (-p) = 0 (the zero polynomial).
6. Closure under scalar multiplication: Multiplying a polynomial by a scalar from 𝔽 results in another polynomial.
7. Associativity of scalar multiplication: a(bp) = (ab)p for all scalars a, b in 𝔽 and all polynomials p.
8. Distributivity of scalar multiplication over vector addition: a(p + q) = ap + aq for all scalars a in 𝔽 and all polynomials p, q.
9. Distributivity of scalar multiplication over field addition: (a + b)p = ap + bp for all scalars a, b in 𝔽 and all polynomials p.
10. Identity element of scalar multiplication: 1p = p for the scalar 1 in 𝔽 and all polynomials p.

Why 𝔽[z] Satisfies the Axioms

• Polynomials are Closed under Addition and Scalar Multiplication: When you add two polynomials or multiply a polynomial by a scalar, the result is always another polynomial with coefficients in 𝔽.
• Associativity, Commutativity, Distributivity: These properties follow directly from the properties of addition and multiplication in the field 𝔽.
• Zero Vector and Additive Inverses: The zero polynomial exists and for every polynomial, there's a corresponding polynomial with the signs of all coefficients flipped, which acts as the additive inverse.

Example (from the original text):

• p(z) = 5z + 1
• q(z) = 2z^2 + z + 1

We can see:

• (p + q)(z) = 2z^2 + 6z + 2 (This is another polynomial)
• (2p)(z) = 10z + 2 (This is another polynomial)

Conclusion

The linearity of any individual polynomial within 𝔽[z] does not impact the fact that 𝔽[z] as a whole satisfies the axioms of a vector space over 𝔽. The vector space structure comes from how the polynomials interact with each other through addition and scalar multiplication, not from the degree of individual polynomials.