Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).
The Attempt at a Solution
I may be complicating things, but let me know if you see something missing.
Doing the appropriate algebra, I manipulated the above expression into (a^2 - 5)^2=23
Expanding the left side, we get a^4 - 10*a^2 + 25 = 23 , i.e. a^4 - 10*a^2 + 2 = 0
So I plan to use f(x)=x^4 - 10*x^2 + 2
From here, I just need to show that it's irreducible.
If it is reducible, there will be either a linear factor or a quadratic factor.
My last step was simply to just use brute force to find a contradiction when comparing the following expressions with my polynomial above:
(ax+b)(cx^3 + dx^2 + ex + f) and