  We prove that the minimizer in the N\'ed\'elec polynomial space of some
degree p > 0 of a discrete minimization problem performs as well as the
continuous minimizer in H(curl), up to a constant that is independent of the
polynomial degree p. The minimization problems are posed for fields defined on
a single non-degenerate tetrahedron in R^3 with polynomial constraints enforced
on the curl of the field and its tangential trace on some faces of the
tetrahedron. This result builds upon [L. Demkowicz, J. Gopalakrishnan, J.
Sch\"oberl SIAM J. Numer. Anal. 47 (2009), 3293--3324] and [M. Costabel, A.
McIntosh, Math. Z. 265 (2010), 297--320] and is a fundamental ingredient to
build polynomial-degree-robust a posteriori error estimators when approximating
the Maxwell equations in several regimes leading to a curl-curl problem.
