PlanetMath: non-isomorphic completions of $\mathbb{Q}$

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non-isomorphic completions of $\mathbb{Q}$

(Theorem)

No field $\mathbb{Q}_p$ of the -adic numbers (-adic rationals) is isomorphic with the field $\mathbb{R}$ of the real numbers.

Proof. Let's assume the existence of a field isomorphism $f:\,\mathbb{R}\to \mathbb{Q}_p$ for some positive prime number . If we denote $f(\sqrt{p}) = a$ , then we obtain

$\displaystyle a^2 = (f(\sqrt{p}))^2 = f((\sqrt{p})^2) = f(p) = p,$

because the isomorphism maps the elements of the prime subfield on themselves. Thus, if $\vert\cdot\vert _p$ is the normed

-adic valuation of $\mathbb{Q}$ and of $\mathbb{Q}_p$ , we get

$\displaystyle \vert a\vert _p = \sqrt{\vert a^2\vert _p} = \sqrt{\vert p\vert _p} = \sqrt{\frac{1}{p}},$

which value is an irrational number as a square root of a non-square rational. But this is impossible, since the value group of the completion $\mathbb{Q}_p$ must be the same as the value group $\vert\mathbb{Q}\setminus\{0\}\vert _p$ which consists of all integer powers of

. So we conclude that there can not exist such an isomorphism.

"non-isomorphic completions of $\mathbb{Q}$ " is owned by pahio.

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See Also: p-adic canonical form

Also defines:

-adic numbers

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Cross-references: powers, integer, value group, value group of the completion, rational, irrational number, prime subfield, maps, isomorphism, prime number, positive, field isomorphism, real numbers, isomorphic, field
There are 2 references to this entry.

This is version 6 of non-isomorphic completions of $\mathbb{Q}$ , born on 2005-01-27, modified 2005-01-28.
Object id is 6671, canonical name is NonIsomorphicCompletionsOfMathbbQ.
Accessed 2121 times total.

Classification:

AMS MSC:	12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
	13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
	13J10 (Commutative rings and algebras :: Topological rings and modules :: Complete rings, completion)
	13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)

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