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I have the following questions that asks for
a)dimension of $Q(i)$ as a Q-vector space and
b)dimension of $C$ as an $R$-vectorspace

Also is the dimension of $R$ as a $Q$-vector space finite?

I think that the dimension for both of them is $2$ because complex numbers can be written as $a+bi$ which means it has $2$ dimensions. For the third part I want to say yes but I can't explain why

I am not sure what it means by 'Q-vector space' or 'R-vector space'. We are using the textbook "Linear Algebra" by Serge Lang and the book does not talk much about what fields are, and so I do not really understand what a field is. (Most definitions I have looked up online of fields involve rings which we have never covered before in class)

Any help would be much appreciated

lllll
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  • have a look here http://math.stackexchange.com/questions/6244/is-there-a-quick-proof-as-to-why-the-vector-space-of-mathbbr-over-mathbb – ir7 Jan 18 '15 at 23:52

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A field, roughly, is any algebraic structure with two laws of composition (addition and multiplication) satisfying the usual alws you know to be true for rational, real and complex numbers. In particular every non zero element has a multiplicative inverse. Other examples of fields are : rational functions over the real or complex numbers (more generally over any field); $\mathbf Z/p\mathbf Z$ for any prime number $p$ — these are finite fields.

$\mathbf R$ is a $\mathbf Q$-vector space, but it is not finite-dimensional, not even countable dimensional, since otherwise, $\mathbf R$ would be countable, which is not true. A basis of $\mathbf R$ over $\mathbf Q$ is called a Hamel basis, but no one has ever seen one. Actually its existence is a consequence of the axiom of choice.

Bernard
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