Question

Find $\mathbf{}\mathbf{[}\mathbf{\mathbb{Q}}\mathbf{\left(}\sqrt[\mathbf{6}]{\mathbf{2}}\mathbf{\right)}\mathbf{:}\mathbf{\mathbb{Q}}\mathbf{]}\mathbf{.}$

Short answer

The value of $[\mathrm{\mathbb{Q}}\left(\sqrt[6]{2}\right):\mathrm{\mathbb{Q}}]$is 6

Step-by-step solution

Describe the Einstein criteria

A polynomial $\mathbf{Q}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{=}{\mathbf{a}}_{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\mathbf{x}\mathbf{+}\mathbf{\cdots }\mathbf{+}{\mathbf{a}}_{\mathbf{n}}{\mathbf{x}}^{\mathbf{n}}$with integer coefficients is said to be irreducible if there exist a prime number p such that the condition given are satisfied.

Find  [ℚ(26):ℚ]..

Consider$[\mathrm{\mathbb{Q}}\left(\sqrt[6]{2}\right):\mathrm{\mathbb{Q}}].$.

Since $\sqrt[6]{2}$is a root of the polynomial role="math" localid="1659161257474" $x^6-2$over$\mathrm{\mathbb{Q}}$ , and$x^6-2$ is irreducible over Einstein criteria.

First, p divides each $a_i$for$i\ne n$ .

Second, p does not divides each$a_n$ .

Third,$p^2$does not divide$a_0$ .

Therefore, $x^6-2$is minimal polynomial of $\sqrt[6]{2}$over$\mathrm{\mathbb{Q}}$ .

This implies,

$[\mathrm{\mathbb{Q}}\left(\sqrt[6]{2}\right):\mathrm{\mathbb{Q}}]=6$

Hence, the value of role="math" localid="1659161332024" $[\mathrm{\mathbb{Q}}\left(\sqrt[6]{2}\right):\mathrm{\mathbb{Q}}]$is 6.