Let $f(x)=2x^3 +5x^2 +7x+6∈Q[x]$. Find a field, smaller than the complex numbers, in which f(x) splits into linear factors.
I know that I should use the theorem that says that "If f(x) is any polynomial over the field F, there is an extension field K of F over which f(x) splits into linear factors". What I am unsure on is how I am supposed to find the field K. And further, how I'm supposed to find the linear factors f(x) breaks down to.
Thank you in advance!
Let $a_1, a_2, a_3 \in \mathbb{C}$ be the roots of $f$. Then, in $\mathbb{C}[x]$ we have that $f(x) = (x-a_1)(x-a_2)(x-a_3)$. Since you are looking for a field that is smaller than $\mathbb{C}$, just take $K = \mathbb{Q}(a_1,a_2,a_3)$ (this is called the splitting field of $f$). Then since $a_1,a_2,a_3 \in K$, we have that $x-a_i \in K[x]$ for $i = 1,2,3$.
In fact, as J.W. Tanner has pointed out, you can get away with only adjoining one root of $f$ to $\mathbb{Q}$. From this, you'll find that $\mathbb{Q}(\sqrt{-7})$ is enough for $f$ to split.
Also, if you are having trouble finding the roots in $\mathbb{C}$, you may first try doing rational roots test. This wont always work, but since $f$ is a degree $3$ polynomial, it will have a real root so that you at least have a chance. In this case, you find that you do get a rational root, and now do long division to find the other factor of $f$.