# Find a basis of E as a vector space over $\mathbb{Q}$

Find a basis for the factor ring $$\frac{\mathbb{Q}}{<16x^4-30x^3+15x^2+6>}$$ as a vector space over $\mathbb{Q}$.

I honestly don't even know how to start this :( I though I would use the Grobner Bases for Ideals stuff in my textbook, but I don't understand where it says "as a vector space over $\mathbb{Q}$" so I'm stuck.

Any help (even just hints) would be greatly appreciated!

Did you mean to write $$\frac{\mathbb{Q}[x]}{<16x^4-30x^3+15x^2+6>}$$ ??
The "as a vector space over $\mathbb Q$" thing means that there is a natural action of $\mathbb Q$ on this ring and that makes the ring a vector space over the field $\mathbb Q$. Every vector space has a basis so you should be able to list elements of the ring which span and are linearly independent (both with respect to $\mathbb Q$).
For the basis, what about the sequence $1, x, x^2, \ldots$? What's the first $x^n$ that can be written (in your ring) in terms of lower powers of $x$? Once you've figured out what $n$ is ask yourself if $1, x, x^2, \ldots, x^{n - 1}$ is a basis.
• In your ring the equation $16x^4−30x^3+15x^2+6 = 0$ holds. This means $1, x^2, x^3, x^4$ are linearly dependent. You can solve this equation for $x^4$. – Jim Mar 11 '14 at 5:10