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Abstract Algebra- Finding the Minimal Polynomial
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[QUOTE="corky23, post: 4349532, member: 472638"] [h2]Homework Statement [/h2] Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).[h2]Homework Equations[/h2] [h2]The Attempt at a Solution[/h2] 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 (ax^2+bx+c)(dx^2+ex+f) [/QUOTE]
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Abstract Algebra- Finding the Minimal Polynomial
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