Class-Field-theory

Question

I am currently on Jürgen Neukirchs Algebraic Numbers theory and have some problems understanding his non-cohomological approach on class field theory. To introduce the notation would cost tremendous amount of time so I am only referring my question to Neukirch-readers. To prove the multiplicativity of his Reciprocity map he formulates two lemmas. For the prove of the second one he uses the "Norm map", which is a homomorphism from the units of the maximal unramified extension of L to the units of the maximal unramified extension of K.(Those elements which are mapped to Zero under a "Henselian valuation" corresponding to a Field are called its units) But since the maximal unramified extension of L is infinte over the base field k, one cannot have a "Henselian valuation" v_K for it because those "Henselian Valuations" are only defined for fields with finite degree over the base field k. So what are the units of a infinite extension L|k?