  We study extensions of piecewise polynomial data prescribed in a patch of
tetrahedra sharing an edge. We show stability in the sense that the minimizers
over piecewise polynomial spaces with prescribed tangential component jumps
across faces and prescribed piecewise curl in elements are subordinate in the
broken energy norm to the minimizers over the broken H(curl) space with the
same prescriptions. Our proofs are constructive and yield constants independent
of the polynomial degree. We then detail the application of this result to the
a posteriori error analysis of the curl-curl problem discretized with
N\'ed\'elec finite elements of arbitrary order. The resulting estimators are
reliable, locally efficient, polynomial-degree-robust, and inexpensive. They
are constructed by a broken patchwise equilibration which, in particular, does
not produce a globally H(curl)-conforming flux. The equilibration is only
related to edge patches and can be realized without solutions of patch problems
by a sweep through tetrahedra around every mesh edge. The error estimates
become guaranteed when the regularity pick-up constant is explicitly known.
Numerical experiments illustrate the theoretical findings.
