Today, in my graduate algebra course, our instructor introduced the notation $\mathbb{Q}(\mu_p)$ as the field of p-th roots of unity.
But, I've been having trouble trying to figure out exactly what elements in this field look like. Is this field analogous to $\mathbb{Q}(\zeta_p)$, where $\zeta_p$ is a primitive pth root of unity ? That is, is $\mathbb{Q}(\mu_p)$ just different notation for $\mathbb{Q}(\zeta_p)$ ?
How can I describe the elements in $\mathbb{Q}(\mu_p)$ as a linear combination of basis elements over $\mathbb{Q}$ ? Does it contain other fields we know about, depending on the value of $p$ ? For example, if $p = 5$, does $\mathbb{Q}(\mu_5)$ contain $\mathbb{Q}(\sqrt{5})$ ?
Thank you for your help, in advance. I really appreciate it.
$\mu_p$ is just a different notation for $\zeta_p$.
$\mu_p$ satisfies an irreducible polynomial of degree $p-1$, so a basis for the field as a vector space over the rationals is given by $\{\,\mu_p^r: r=0,1,\dots,p-2\,\}$.
The field contains lots of other fields that you may or may not know about, some of which you may learn about as the course goes on. It is not trivial to prove that it contains ${\bf Q}(\sqrt p)$ if $p\equiv1\bmod4$, and ${\bf Q}(\sqrt{-p})$ if $p\equiv-1\bmod4$.