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February 25

Expressing solution of a cubic in radicals

Hi. I'm looking for the unique positive real root of a certain 3rd degree polynomial with coefficients in $\mathbb {Q} \left[{\sqrt {13}}\right]$ . This should be expressible in radicals, per history, if I'm not mistaken.

I've solved it, following Cardano's method in our article, cubic function, but I'm having trouble expressing the solution in terms of radicals of real numbers. I can get it in terms of radicals of complex numbers, or in purely real terms with a cosine.

The function is $f(x)=x^{3}+3{\sqrt {13}}\cdot x^{2}+12x-16{\sqrt {13}}$ , and the solution I've got is:

$x={\frac {1}{81}}\left[{\overline {z}}z^{\frac {2}{3}}+81z^{\frac {1}{3}}-81{\sqrt {13}}\right]$ , with $z={\sqrt {13}}+2i{\sqrt {179}}$ .

Or, equivalently,

$x=6\cos \left({\frac {\alpha }{3}}\right)-{\sqrt {13}}$ , where $\alpha$ is an angle in the first quadrant with $\tan(\alpha )={\frac {2{\sqrt {179}}}{\sqrt {13}}}$ .

Is this the best I can do, or is there some way to massage the first expression into one involving only reals and radicals? I could look at the equation expressing $\cos \left(\alpha \right)$ as a cubic function of $\cos \left({\frac {\alpha }{3}}\right)$ , but I think that'll take me in a circle.

What I really want to know is which field extension of $\mathbb {Q}$ the solution is in. I mean, I'm pretty sure it's in $\mathbb {Q} \left[{\sqrt {13}}\right]\left[z^{\frac {1}{3}}\right]$ , but is there any way to find a real extension that it's in? Will ${\sqrt {179}}$ do it? Any ideas are appreciated; thanks in advance. -GTBacchus^(talk) 03:29, 25 February 2010 (UTC)[reply]

It should be a 6th degree extension so I'm not sure how sqrt(179) is supposed to help. I vaguely remember figuring out a messy way to do a hw problem like this with a bunch of linear algebra and then being told I had done a lot more calculation than was really needed. The article Minimal polynomial (field theory) doesn't look too helpful, unfortunately. The Mathworld reference at that article does mention that Mathematica can figure it out for you. 75.62.109.146 (talk) 03:49, 25 February 2010 (UTC)[reply]

I'm at least as interested in the journey as in the destination in this case, so simply getting the answer is only a partial solution. I realize that the splitting field of the polynomial is degree 6 over Q, but that doesn't mean that our particular x is a degree 6 number, right? I mean, the cube root of 2 satisfies x^6-4=0, but its degree as an algebraic number is only 3. -GTBacchus^(talk) 04:30, 25 February 2010 (UTC)[reply]

x^6-4 = (x^3+2)(x^3-2) and the splitting fields of those two factors are the same, so the overall splitting field still has degree 3. This is what you would expect--the degree of the splitting field is the degree of the minimal polynomial, which for the cube root of 2 is x^3-2. (edited) 75.62.109.146 (talk) 06:26, 25 February 2010 (UTC)[reply]

The degree of the splitting field of x^3-2 over Q is 6, not 3 - see splitting field. The field obtained by adjoining the real cube root of 2 to Q does have degree 3 over Q, but it is not a splitting field - you have to adjoin a primitive cube root of 1 as well to get a splitting field. In general, the splitting field of an irreducible polynomial of degree n may have a degree as large as n factorial.Gandalf61 (talk) 09:54, 25 February 2010 (UTC)[reply]

Thanks, I must have been confused. I had a diagram in mind involving adjoining one of the complex cube roots of 2 and getting the rest from there, but it was wrong. 75.62.109.146 (talk) 17:52, 25 February 2010 (UTC)[reply]

Well, you can plug the coefficients into the first of these formulae. There is undoubtedly a better way, though! (If you just want the answer and don't care about how you get it, any decent maths computer package will be able to get it for you.) --Tango (talk) 03:54, 25 February 2010 (UTC)[reply]

No, I'm interested in how to get there. As for those formulas you linked to, I tried that. The quantity under the square root comes out negative, and we end up with the first expression for x that I've got above. -GTBacchus^(talk) 04:29, 25 February 2010 (UTC)[reply]

Try one of the other formulae, then. For one of them, the i's should cancel out since any cubic with real coefficients will have a real root. --Tango (talk) 18:13, 25 February 2010 (UTC)[reply]

You see, that's exactly what my problem is. I know that the function has 3 distinct real roots, and those formulas all have imaginary numbers in them. I know that the imaginary parts must therefore cancel, but I can't seem to make it happen, unless I use trig functions. If I use the "evalf" command on Maple, I get a real number, but no indication of how Maple got there. (For "evalf", it's probably numerical methods anyway.)

My question is: can I algebraically eliminate the imaginary numbers in my first expression above, and if so, how? If expressing the answer in cosines is the best I can do, I accept that, but I'd like to know why. If, on the other hand, there is a way to actually eliminate the imaginary parts, then I'm asking for help to find that way. Is this number expressible in terms of radicals of real numbers? Does that make sense? -GTBacchus^(talk) 18:46, 26 February 2010 (UTC)[reply]

OK, so the base field

\mathbb {Q} ({\sqrt {13}})

is formally real, the cubic is irreducible, and it has three real roots. Unless I'm missing something, this is exactly the situation of casus irreducibilis, hence the answer is no, it's not expressible using only real radicals.—Emil J. 19:03, 26 February 2010 (UTC)[reply]

Aha. In that case, I don't feel dumb for being unable to eliminate the i's. Thank you; that's an interesting link. -GTBacchus^(talk) 01:11, 27 February 2010 (UTC)[reply]

The mathworld link points to this which has a few examples of doing it by hand. It looks plausible that you can do your example with the same methods. 75.62.109.146 (talk) 03:55, 25 February 2010 (UTC)[reply]

I'll try this and report back whether it works. :) -GTBacchus^(talk) 04:29, 25 February 2010 (UTC)[reply]

Related previous question: [1]. If you're able to express the solution in radicals involving reals only, I (User_talk:NorwegianBlue) would love to hear about it! --85.200.133.168 (talk) 20:46, 26 February 2010 (UTC)[reply]

The casus irreducibilis of expressing the real solutions of the cubic equation using reals only, gave historically rise to the invention of the complex numbers. Bo Jacoby (talk) 20:28, 28 February 2010 (UTC).[reply]

Involving Euclidean algorithm for polynomials

I'm reading: "Assume GCD(r,s) = 1. Then the Euclidean algorithm gives polynomials B(t) and C(t) such that: $B(t)(1+t+\dots +t^{r-1})+C(t)(1+t+\dots +t^{s-1})=1$ ." This is outside my usual area, and I'm not sure how to interpret "the Euclidean algorithm gives". Do they mean that it's obvious for some reason that $1+\dots +t^{r-1}$ and $1+\dots +t^{s-1}$ have GCD = 1, and so the Euclidean algorithm (specifically Bezout's identity) gives the linear combination to 1? Or do they mean that you could (omitting the details) actually do the Euclidean algorithm by repeated long-divisions and you'd discover thereby that the GCD is 1?

In either case I'm missing part of the argument. I don't know why those two polynomials have GCD 1 (though maybe it's an easy fact that I've forgotten), and I also don't know how to use the Euclidean algorithm to show it- I can do a step or two but things get too messy.

I'd also like to know exactly what the polynomials B and C are. They should be expressible in terms of s and r, right? Like I said I get lost in the details when I try to construct them by the Euclidean algorithm. Thanks- Staecker (talk) 14:38, 25 February 2010 (UTC)[reply]

Think of for instance with r 3 and s 7, then we can get numbers so 3r-7s=1 and in fact 3.5-2.7 works. now think of those polynomials as strings of 1s: 111 111 111 111 111 - 1111111 1111111 0 = 1. You can translate these directly into B and C by the groupings, B is

t^{12}+t^{9}+t^{6}+t^{3}+1

and C is

-(t^{8}+t)

. Don't know if that's what they mean but it shows how you can get them easily when the GCD is 1. Dmcq (talk) 15:16, 25 February 2010 (UTC)[reply]

(edit conflict) Since

t^{n}-1=(t-1)(1+t+\dots +t^{n-1})

, the statement that your two polynomials have GCD 1 reduces to the statement that

t^{s}-1

and

t^{r}-1

have no root save 1 in common, i.e. that a number that is both an rth root of unity and an sth root of unity is a 1st root of unity, and this follows easily from r and s being coprime. Algebraist 15:24, 25 February 2010 (UTC)[reply]

Thanks a lot- two great answers. Just what I needed (I hope). Staecker (talk) 16:31, 25 February 2010 (UTC)[reply]