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In this expository paper we provide a geometric proof of the local LanglandsCorrespondence for the groups $\operatorname{GL}_{1}$ defined over $p$-adicfields $K$. We do this by redeveloping the theory of proalgebraic groups anduse this to derive local class field theory in the style of Serre andHazewinkel. In particular, we show that the local class field theory of Serreand Hazewinkel is valid for both equal characteristic and mixed characteristicultrametric local fields. Finally, we use this to prove an equivalence of thecategories of smooth representations of $K^{\ast}$ with continuousrepresentations of $W_K^{\text{Ab}}$ in order to deduce the Local LanglandsCorrespondence for $\operatorname{GL}_{1,K}$.

Serre-Hazewinkel Local Class Field Theory and a Geometric Proof of the Local Langlands Correspondence for $\operatorname{GL}(1)$