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Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

 

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, here) Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer. Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, here) Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer. Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, NOW) Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer. Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, here) Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer. Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

Part 2: a family of $p$-adic fields. Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer. Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.

Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}_p$:

$$F_v\simeq K ?$$

In imprecise words, does any $p$-adic field come as a completion of a number field?

Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, NOW) Let $(K_p)_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer. Does there exist a condition on $(K_p)_p$ such that there is one number field $K$ recovering each $K_p$ as a completion?