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What are radicals?
By Olof Sisask (P3033) on Saturday, December 2, 2000 - 06:25 pm:

What are radicals, with respect to statements such as "the general algebraic equation of the 5th degree cannot be solved in radicals"?

Thanks,
Olof.

By Dan Goodman (Dfmg2) on Saturday, December 2, 2000 - 06:30 pm:

A radical is something like sqrt(5), sqrt(1+sqrt(5)) or more complicated like cuberoot(1+fifthroot(7+2×twentiethroot(42-sqrt(2)))). The constant terms have to be rational numbers. To say "the general algebraic equation of the 5th degree cannot be solved in radicals" means that there are quintics whose roots can't be written as radicals.

By James Lingard (Jchl2) on Saturday, December 2, 2000 - 06:31 pm:

They're expressions involving only integers, plus the operations of addition, subtraction, multiplication, division and the extraction of nth roots for any integer n.

So sqrt(3 - sqrt(5/2))/3 is a radical, but p is not.

James.

By Vivien Easson (Vre20) on Monday, December 4, 2000 - 10:24 am:

Dear Olof,

I think that an example of a quintic (polynomial of degree 5) which cannot be solved by radicals is the following:
f(X) = X5 - X - 1

The area of mathematics which deals with questions like this is called Galois Theory. Roughly, to every polynomial you associate a group (its Galois group) of those symmetries of its roots in the complex plane which fix the rational numbers. The polynomial above has Galois group S[5], which means that every permutation of roots fixes the rational numbers.
See Ian Stewart's book called Galois Theory for more details and some history. I find this area of maths very beautiful.

Best wishes, Vivien.

By Olof Sisask (P3033) on Monday, December 4, 2000 - 05:24 pm:

Thanks guys!

So that's what Galois Theory is. I'll be sure to check out the book you mentioned Vivien.

Cheers,
Olof.