Is there a quick proof as to why the vector space of $\\mathbb{R}$ over $\\mathbb{Q}$ is infinite-dimensional?

Question

It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?

Answer 1

To establish the infinitude of the dimension of R as a vector space over Q, we are asked to prove the existence of a infinite set of real numbers that are linearly independent over Q. There are a few ways to prove this. One is to use the transcendental number e or pi. Say, we use e, then the set of all powers of e is an infinite set of real numbers that are linearly independent over Q. For if not, then e satisfies a polynomial equation which implies that e is algebraic over Q, contradicting the fact that e is transcendental over Q.

Another construction is to add certain roots to Q repeatedly. For example,, we can add the square roots of p for all prime numbers p. Each time the dimension of the extension over Q will increase and the end result is an infinite dimensional extension over Q, which is merely a subfield of R, and hence an infinite dimensional vector space over Q. Then so is R. There are many ways to construct such a set, though.

Answer 2

Consider any finite set S of linearly independent vectors in R over Q. Then the span of S over Q is countable and thus not all of R.