The numbers $e^{2r\pi i/3} + \sqrt[3]{2}e^{2s\pi i/3}$, with $r,s \in\{0,1,2\}$, all satisfy that equation. So that gives you the nine roots of $f(x)$, of which $\alpha$ is one. I'm guessing that $f(x)$ is irreducible over $\mathbb{Q}$ but I don't see how to prove that.kalish said:I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root.
How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?
A minimum polynomial is the smallest degree monic polynomial with integer coefficients that has a given algebraic number, $\alpha$, as a root. In other words, it is the polynomial of lowest degree that satisfies the equation $\alpha^n + a_{n-1}\alpha^{n-1} + ... + a_1\alpha + a_0 = 0$, where the coefficients $a_0, a_1, ..., a_{n-1}$ are all integers.
Finding the minimum polynomial allows us to express $\alpha$ as a polynomial with rational coefficients, which is useful for simplifying expressions and solving equations involving $\alpha$. It also helps in determining the degree of the field extension $\mathbb{Q}(\alpha)$, which is important in algebraic number theory.
To find the minimum polynomial, we can start by considering the powers of $\alpha$ up to the degree of the field extension $\mathbb{Q}(\alpha)$. Then, we can use the fact that $\alpha$ satisfies a monic polynomial of degree $n$ to write out a system of equations with $n$ unknowns, in this case the coefficients of the minimum polynomial. Solving this system of equations will give us the minimum polynomial of $\alpha$.
No, the minimum polynomial must have integer coefficients. This is because $\alpha$ is an algebraic number, meaning it is a root of a polynomial with integer coefficients. Since the minimum polynomial of $\alpha$ is the smallest such polynomial, it must also have integer coefficients.
There are a few methods that can be used to find the minimum polynomial of $\alpha$ over $\mathbb{Q}$. These include using the rational root theorem, checking for irreducibility using Eisenstein's criterion, and using the minimal polynomial algorithm. However, there is no one-size-fits-all shortcut and the method used may vary depending on the specific algebraic number and its properties.