In Neukirch’s book “Algebraic Number Theory”, Proposition II.5.7, the following is insisted: for a mixed characteristic local field $K$ with a residue field $\mathbb{F}_q$, $q = p^f$, then one has an isomorphism $K^\times \cong \mathbb{Z} \times \mathbb{Z}/(q-1)\mathbb{Z} \times \mathbb{Z}/p^a\mathbb{Z} \times \mathbb{Z}_p^{[K:\mathbb{Q}_p]}$ for some $a$. enter image description here

I agree to the following statements: algebraically and topologically, $K^\times \cong \mathbb{Z} \times \mathbb{Z}/(q-1)\mathbb{Z} \times U^{(1)}$; algebraically $U^{(1)} \cong \mathbb{Z}/p^a\mathbb{Z} \times \mathbb{Z}_p^{[K:\mathbb{Q}_p]}$ for some $a$. However I don’t have any idea to prove that $U^{(1)}$ is topologically isomorphic to the right hand side. Does anyone have good idea?