Can these equations be solved by radicals?

[Pan Miroslav]

Hello, the question is like this: Study if the following equations can be solved by radicals and, if it is possible, find a radical sequence to solve them
1) x^6 - 3x^2 - 2 = 0
2) x^6 + 2x^5 - 5x^4 - 5x^3 - 5x^2 + 2x + 1 = 0

(this is my first question here and I'm not sure if I can use tex here and if yes, how can I use it)

My tries:
1) I know that x^6 - 3x^2 - 2 = (x^2 - 2)(x^2 + 1)^2, so the roots are 2^1/2, (-2)^1/2, i, -i. So I guess the first one can be solved by radicals, its radical extension is Q(2^1/2, i) and its radical sequence is 2^1/2, i. Is this right? I'm not doing any progress for second one yet, because I don't know if this approach is even right one.

Thanks for help.

 

[stapel]

Hello, the question is like this: Study if the following equations can be solved by radicals and, if it is possible, find a radical sequence to solve them
1) x^6 - 3x^2 - 2 = 0
2) x^6 + 2x^5 - 5x^4 - 5x^3 - 5x^2 + 2x + 1 = 0

(this is my first question here and I'm not sure if I can use tex here and if yes, how can I use it)

My tries:
1) I know that x^6 - 3x^2 - 2 = (x^2 - 2)(x^2 + 1)^2, so the roots are 2^1/2, (-2)^1/2, i, -i. So I guess the first one can be solved by radicals, its radical extension is Q(2^1/2, i) and its radical sequence is 2^1/2, i. Is this right? I'm not doing any progress for second one yet, because I don't know if this approach is even right one.

Wolfram Alpha shows the factorization and solutions for the second equation:

https://www.wolframalpha.com/input/?i=x^6+++2x^5+-+5x^4+-+5x^3+-+5x^2+++2x+++1+=+0

But I don't know how you'd been expected to "find" these on your own. :shock:

 

[Pan Miroslav]

Well I can find the roots even for the second polynomial (it is reciprocal, so it is not that hard), but I am asked to solve it using radical sequence, which I don't even basically now how is supposed to be done.