Question

$x3 + x + 1 is cubic$
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Answer 1

Answer:Since f(x)=x3−3x−1 is cubic, if it can be reduced, it will have a linear factor, however by the rational root test, ±1 are not roots, hence f is irreducible over Q.

disc(f(x))=−4(−3)3−27(−1)2=4(27)−27=34=92 Since this cubic has a square discriminant, we have Gal(L/K)=A3 and hence [L:K]=3

We see that over F5, f(x)=x3−3x−1 has f(0)=4,f(1)=2,f(2)=1,f(3)=2,f(4)=1 so that f is irreducible over F5.

Now F5 is a finite field, so is perfect and any algebraic extensions shall be perfect.

(WE CAN EMBED Gal(L2/F5) INTO Gal(L/Q) not sure how to write this rigorously)

Step-by-step explanation: