Solve 4th Order Polynomial w/Integer Coefficients: Algebraic Int

[ballzac]

The rest of this sheet of problems was a piece of cake, and I think this is meant to be one of the easier problems on it, but I'm not quite sure how to do it.

Homework Statement

Find a fourth order polynomial with integer coefficients for which $$1+\sqrt{5}-2\sqrt{3}$$

The Attempt at a Solution

I tried rearranging it thus
$$(x-1)^2=17-4\sqrt{15}$$
wasn't sure what to do here, but fourth order is required, so I tried squaring both sides again...
$$(x-1)^4=529-8\sqrt{15}$$
Now I don't know how to get rid of the surd. I tried expanding out the left hand side, which didn't help. Hopefully it's not my arithmetic, as that would be a little embarrassing, but I have tried several times and not resolved it. Any help pointing me in the right direction would be greatly appreciated.

Hmm...I just realized that if $$\sqrt{15}$$ is rational then I can multiply the whole thing by the denominator. So I will try to figure that out if it is rational, however my intuition says it is not (my intuition has failed me in the past, so I will try to prove it), and it may be difficult to find what it is as a ratio if it is rational.

 

[epenguin]

You just need to have the square root on its own on one side.
I don't think you don't know how to do that, more likely you have forgotten.

If you can't remember or work it out don't be frightened or ashamed to go back to your more elementary schoolbooks, general advice.

 

[Hurkyl]

Alternatively... you have two equations, and you're trying to eliminate the term involving $\sqrt{15}$. That's something else you know how to do...

 

[ballzac]

You just need to have the square root on its own on one side.
I don't think you don't know how to do that, more likely you have forgotten.

If you can't remember or work it out don't be frightened or ashamed to go back to your more elementary schoolbooks, general advice.

Oh my God. I feel a tad stupid now. I knew it would be something simple. I was right that it's the easiest question on the sheet. Thanks for the help.