The following question has already been asked by the OP at https://math.stackexchange.com/questions/3454735/on-self-duality-of-non-archimedean-local-fields. Due to a lack of feedback, the OP was forced to ask the same question in the hope of getting better feedback.

To let $ K $ to be a non-Archimedean local field. His additive group $ K ^ + $ is a locally compact abelian Hausdorff group. My question is this:

is $ K ^ + $ isomorphic to its pontryagin dual $ forbid {K ^ +} $ as a topological group?

**Remarks:**

1) Fix a nontrivial unitary character $ psi: K ^ + rightarrow mathbb {C} ^ { times} $, For each $ a in K $, the map $ a psi: K ^ + rightarrow mathbb {C} ^ { times}, ; x mapsto psi (ax) $ is a unified character of $ K ^ + $and the card $ a mapsto a psi $ is an isomorphism of abstract groups $ K ^ + $ on to $ forbid {K ^ +} $ (see *e.g.* (1, Section 1.17 Prop.)). Unfortunately, the argument in (1, Section 1.17 Prop.) Does not prove (as far as I can see) that the previous map is a homeomorphism.

2) Leave $ p in mathbb {N} $ be the feature of the residual field of $ K $, It is known that as a topological field, $ K $ is either isomorphic to a finite extent $ mathbb {Q} _p $ or to an area of the formal Laurent series $ mathbb {F} _q ((t)) $ from where $ q $ is a power of $ p $, Since the additive group of $ mathbb {Q} _p $ It is known that it is (as a topological group) isomorphic to its pontryagin – dual (cf. *e.g.* In the introductory paragraphs in (2), the question can be reduced to the case in which $ K = mathbb {F} _q ((t)) $, But I can not find clues or arguments that indicate a positive answer.

Thank you in advance.

**references:**

(1) C.J. BUSHNELL AND G. HENNIART, *The local Langlands conjecture for GL (2)*, Springer, 2006.

(2) L. CORWIN, *Some remarks on locally compact abelian groups that double themselves*, Trans. Amer. Mathematics. Soc. **148** (1970), 613? 622nd