Wednesday, May 8, 2019

abstract algebra - Degree of splitting field extensions

The problem states:




Let f(x)=x3+px+q be an irreducible cubic polynomial with rational coefficients
and let K be the splitting field of f(x) over Q. Prove that [K:Q]=3 if and only if 4p327q2 is a square in Q.




Here is how far I've come on the problem. Since f is cubic and the degree of the extension is 3, then non of the roots are in Q hence they are in K. Now, I am having trouble connecting this piece of information with the given value being a square in Q.



Thanks in advance.

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