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Vector space has dimension less than d

1. The problem statement, all variables and given/known data
Problem given to me for an assignment in a math course. Haven't learnt about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below.


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2. Relevant equations
Unsure of relevant equations

3. The attempt at a solution
so far i am just trying to understand what the question is asking me to do, i am showing that the vector space V, which is just the rational numbers over the field Q? has dimension strictly less than d, where d is >= 1?
 

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The vector space ##V## is spanned by ##d## elements. Therefore, its dimension is at most ##d##. The question asks you to show that the generators are not linearly independent, which means you can remove one of the generators of ##V## such that the remaining generators still span ##V##. It will then follow that the soace has dimension ##d-1##.

The question gives a hint how to show that the generators are not linearly independent.

Also, the space is not the rational numbers.

It are ##\mathbb{Q}##-linear combinations of powers of roots of unity.
 

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