Existence of irreducible polynomials over Q yielding injective evaluation on M2(Q)

Ascertain whether there exists an irreducible polynomial f in Q[x] of degree greater than 2 such that the evaluation map eva_{f, M_2(Q)}: M_2(Q) -> M_2(Q), A -> f(A), is injective.

Background

Remark ‘remark2’ presents a concrete example over Q where injectivity fails in dimension n = 2 despite h(x) being irreducible of degree d = 3, illustrating that non-injectivity can occur for n < d. This motivates seeking positive examples.

The open question asks if there exists any irreducible polynomial over Q with degree greater than 2 that induces an injective evaluation map on M_2(Q), thereby probing whether injectivity is ever attainable in this setting.

References

From Remark~\ref{remark2}, we can also formulate another open problem: Is there an irreducible polynomial $f$ over $\mathbb{Q}$ of degree greater than $2$ such that the evaluation map $\mathrm{eva}_{f,\mathrm{M}_2(\mathbb{Q})}$ is injective?

— On the injectivity of evaluation maps induced by polynomials on certain algebras (2508.17570 - Kutzschebauch et al., 25 Aug 2025) in Problem (Open) after Remark ‘remark2’, Section “Matrix algebras over fields”