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I came across this problem in Halmos, Finite-Dimensional Vector Spaces, page 16.

Is the set R of all real numbers a finite-dimensional vector space over the field Q of all rational numbers. There is a reference to a previous example which says that with the usual rules of addition and multiplication by a rational R becomes a rational vector space. My answer to the question would be that R is not a finite-dimensional vector space over the field Q.

The author goes on to say that the question is not trivial and it helps to know something about cardinal numbers.

Can anyone please expand on this.

Thanks Matheinste.