On Wednesday, April 30, 2014 1:07:55 AM UTC-7, John Cremona wrote:

> NO!  If f1 and f2 define the same extension field then <f1,f2> will 
> not be prime, e.g. f1=x^2-2 and f2=x^2-8 over QQ. 
>

you probably mean f1=x^2-2 and f2=y^2-8 in QQ[x,y] .  <x^2-2,x^2-8>=<1> is 
also not prime but that doesn't have much to do which fields the 
polynomials define.

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