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.

  • $\begingroup$ I'm confused about the "lower powers of x." I don't know if this means a combination of several x's or what? Sorry. I just don't understand what I'm supposed to do here at all $\endgroup$ – user132039 Mar 11 '14 at 3:31
  • $\begingroup$ 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 $x^4$. $\endgroup$ – Jim Mar 11 '14 at 5:10
  • $\begingroup$ How do I know that I will solve the equation for x^4? Do I always solve for the variable with the highest exponent? $\endgroup$ – user132039 Mar 11 '14 at 6:06
  • 1
    $\begingroup$ You need to figure out a spanning set, so you need to figure out which vectors can be written as other vectors. Eliminating the highest degree (when writing some vectors in terms of others) means the process must stop because the degree is always decreasing and can't decrease below zero. $\endgroup$ – Jim Mar 11 '14 at 6:40

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