what is the dimension of the vector space of real numbers over the field of rational numbers? and why?

Question

what is the dimension of the vector space of real numbers over the field of rational numbers? and why?

Answer 1

The real numbers are uncountable. The rational numbers are countable. It is a fact that the direct sum of countable sets is countable, so the real numbers cannot be a countable direct sum of copies of the rational numbers. That is to say, the real numbers have uncountably infinite degree over the rational numbers.

Answer 2

rational numbers algebraic numbers (over the rationals) p-adic numbers fields of finite characteristic those are possibly the main undemanding edit: specific, Q^3 is a vector area over the rational numbers. R and R^3 are additionally vector areas over the rational numbers (in spite of the undeniable fact that countless-dimensional). that's accomplished by way of figuring out a vector in R^3 with the three approximating sequences of rational numbers for each element

Answer 3

All finite extensions are algebraic. Since R/Q is not algebraic, it is not finite.



Another way to see this:



If you were to add the √p for all prime p to Q, you would not get all of R, but would create an infinite extension. Therefore, even the algebraic closure of Q intersect R is infinite over Q.

Answer 4

infinity,

since you need an infinite number of generators,

as many as the irrational numbers (not all)