Vector space has dimension less than d

[UOAMCBURGER]

Homework Statement

Problem given to me for an assignment in a math course. Haven't learned 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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Homework Equations

Unsure of relevant equations

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?[/B]

 

[member 587159]

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.