Determine the Galois group of $x^3 + 3x^2 - 1$ over $\mathbb{Q}$

Substituting $x=y-1$ for $x$ yields the well-known equation $y^3-3y+1=0$, whose roots are $2cos(2\pi n/9)$ for $n=1, 4, 7$. This equation has Galois Group $A_3$; in fact, if a root is $\alpha$, then the other two roots are $\alpha ^2-2$ and $2-\alpha-\alpha ^2$, showing that adding $\alpha$ to the rationals also adds the other two roots.

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Abstract Algebra

Galois Theory

Galois Extensions

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