Roots of $x^4-5x+5$

Question

Suppose $z$ is a complex root of $x^4-5x+5$. What is the extension degree of $\mathbb{Q}(z):\mathbb{Q}$? I suspect it is 4 but I don't have any strategy how to prove it.

Answer 1

The degree is equal to the degree of the minimal polynomial of $z$ (which has rational coefficients). Since your polynomial is irreducible and with rational coefficients, it must be the minimal polynomial of $z$. Therefore, the degree of the extension is $4$.

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To prove that the degree of the minimal polynomial is indeed 4, you proceed as follows:

Suppose that the minimal $Q$ polynomial has degree less than $4$. Then, denoting by $P$ the initial polynomial, we know that there exist polynomials $S,R$ with rational coefficients such that $deg R < deg Q$ with

$$P=SQ+R $$

If $R$ is non-zero it follows that $R(z)=0$ and we have found a polynomial with rational coefficients, which has $z$ as a root and has a degree less than the minimal polynomial. This is a contradiction.

If $R=0$ then $Q|P$ and this is not possible, since $P$ is irreducible.