Are there number fields with 3, 5, or 7 (and so on) real embeddings on a number field $L$ into $\bar{\mathbb{Q}}$ where $[L:\mathbb{Q}]=2k+1$? I am trying to find the possible subgroups of the unit group in the ring of integers, $\mathcal{O}_L$. While using Dirichlet’a unit theorem, I started to doubt myself whether such an extension is possible. It is clear to me that complex embeddings come in pair, but I am not sure if I can have a number field, say of degree 3, where all the extensions are real.
I would appreciate any hint or examples.
Thanks!!
 A: Argument borrowed from a related question.
Let $n$ be greater than $1$. Then by Dirichlet's theorem on arithmetic progressions, there exists a smallest prime $p(n)$ such that $n$ divides $p(n)-1$.
Now consider the cyclotomic extension $\mathbb{Q}(\zeta_{p(n)})$. We know that the Galois group $G$ of $\mathbb{Q}(\zeta_{p(n)})$ over $\mathbb{Q}$ is cyclic of order $p(n)-1$. Let $H$ be the subgroup of order $\tfrac{p(n)-1}{n}$. Then the fixed field $K_n\stackrel{\text{def}}{=}\mathbb{Q}(\zeta_{p(n)})^H$ has degree $n$ and Galois group $G/H$ cyclic of order $n$.
But Galois extensions of odd degree are totally real. Therefore if $n$ is odd, then $K_n$ has exactly $n$ real embeddings.
Here are minimal defining polynomials for the first few odd $n$:


*

*$x^3 - x^2 - 2x + 1$ (LMFDB 3.3.49.1)

*$x^5 - x^4 - 4x^3 + 3x^2 + 3x - 1$ (LMFDB 5.5.14641.1)

*$x^7 - x^6 - 12x^5 + 7x^4 + 28x^3 - 14x^2 - 9x - 1$ (LMFDB 7.7.594823321.1)

*$x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5x - 1$ (LMFDB 9.9.16983563041.1)

A: 
Given a number field of degree $n$ then the number of real embeddings and pairs of complex embeddings $r_1,r_2$ can be anything prior that $r_1+2r_2 = n$.

Let $f\in \Bbb{Z}[x]_{monic}$ be irreducible in $\Bbb{F}_p[x]$, let $g \in \Bbb{Z}[x]$ of same degree with $r_2$ pairs of complex roots and with $r_1$ distinct real roots, then $f+mp g$ is irreducible and since $g+\frac{f}{mp} \to g$ locally uniformly, for $m$ large enough it has $r_1$ real roots and $r_2$ pairs of complex roots. With $\alpha$ one of its roots then $\Bbb{Q}(\alpha)$ has $r_1$ real embeddings and $r_2$ pairs of complex embeddings.
