# 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!!

• $x^3-2x+1$ is irreducible, it has a real root $\alpha_1$, its discriminant $D=-4(-2)^3-27 \ 1^2= 5 = (\prod_{i < j} (\alpha_i-\alpha_j))^2$, its splitting field is $\Bbb{Q}(\alpha_1,\sqrt{D}) \subset \Bbb{R}$ thus the cubic field $\Bbb{Q}(\alpha_1)$ has 3 real embeddings. For $x^3-3x+1$ it is $D = 81$ thus $\Bbb{Q}(\alpha_1)$ is Galois and totally real. Aug 14, 2019 at 21:47
• See this thread. Close to being a duplicate of that actually. Aug 15, 2019 at 4:42

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)

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.