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Let $V$ be set of real numbers and $K$ the field of rational numbers.

Is $V$ a vector space over $K$, with ordinary addition of real numbers and multiplication by rational numbers?

egreg
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kay
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  • Similar question: http://math.stackexchange.com/questions/257504/the-real-numbers-form-a-vector-space-over-the-rationals-i-e-with-q-as-the-scal – N.U. May 19 '13 at 14:50

3 Answers3

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Yes, $V$ is a vector space over $K$, but it is of infinite dimension over $K$. Choosing a basis for $V$ over $K$ is a non-trivial matter that requires non-constructive foundational material such as the axiom of choice, see http://en.wikipedia.org/wiki/Axiom_of_choice

Mikhail Katz
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  • Mikhail, would the opposite be the case as well, i.e., can the Rationals be made into a vector space over the Reals? Well, as a field, they are a vector space over themselves. I'm a bit stale on this from long back, but maybe we can use Realification, and, over the Complexes, maybe Complexification? – MSIS Sep 27 '22 at 18:04
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A vector space over a field $K$ is any set $V$ equipped with functions $V\times V\to V$ and $K\times V\to V$ satisfying the axioms (I sometimes call them "computation rules").

So, since $\mathbb{R}$ with the two functions you mention satisfies the axioms, it is a vector space over the rationals. It is infinite dimensional, by reasons of cardinality: a finite dimensional vector space over an infinite field $K$ has the same cardinality as $K$.

In the same vein, $\mathbb{C}$ is a vector space over $\mathbb{Q}$ as well. This has important applications in the theory of algebraic numbers, because one can use the powerful machinery of vector spaces. Note that $\mathbb{C}$ and $\mathbb{R}$ are vector spaces over any of their subfields. One of the most useful results in this respect is the formula

$$ [M:L][L:K]=[M:K] $$

where $M$, $L$ and $K$ are subfields of $\mathbb{C}$ (any field would do, actually) with $K\subseteq L\subseteq M$ and with $[L:K]$ we denote the dimension of $L$ as a vector space over $K$.

egreg
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Anytime you have a field extension $V/K$ you can consider $V$ as a vector space over $K$.

Edoardo Lanari
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