A number field $K$ is said to have a *power basis* if there is an $\alpha \in K$ such that the full ring of integers $O\_K$ is the $\mathbb{Z}$-linear span of $1,\alpha,\alpha^2,\ldots,\alpha^{\deg{K}-1}$. In other words, $O\_K = \mathbb{Z}[\alpha]$; another term for this is monogenic. This happens for example for the quadratic and the cyclotomic fields. The monogenic number fields are presumably very rare; results of Bhargava imply that they are a negligible fraction (zero density) among the number fields of degree $3, 4$, or $5$, and this is conjectured to hold in every fixed degree $d > 2$.


It is easily seen that totally $p$-adic number fields of degree $d$ are never monogenic for $d \gg p$. (To illustrate this, consider that the split primes in the cyclotomic field of level $N$ are the ones $\equiv 1 \mod{N}$; so they are in particular $> N > \phi(N)$.) 


**Question.** *Are there monogenic totally real number fields of arbitrarily high degree?*


To put it differently: for totally real algebraic integers of arbitrarily high degree, may the ring $\mathbb{Z}[\alpha]$ be integrally closed?


Dummit and Kisilevsky [*Indices in cyclic cubic fields*, in "Number Theory and Algebra," 1977] have shown that infinitely many totally real cubic fields have a power basis. There exist totally real monogenic sextic fields; an example, taken at will from page 116 of [I. Gaal, *Diophantine Equations and Power Integral Bases: New Computational Methods*] is the field generated by a root of
$$
X^6 - 5X^5 +2X^4 +18X^3 - 11X^2 -19X+1.
$$
Are there examples of higher degree?



Can there be a power basis for a totally real field of high degree?
[Kedlaya](http://arxiv.org/abs/1103.5728) proves that there are infinitely many irreducible integer polynomials of every degree with real roots and square free discriminant; if $P$ has square free discriminant then $\mathbb{Z}[x]/P(x)$ is integrally closed. More precisely, he constructs $c N^{1/(n-1)}$ different fields of this sort with discriminant in $[-N,N]$.


According to [Ash-Brakenhoff-Zarrabi](https://www2.bc.edu/~ashav/Papers/ABZ-proofs.pdf), Hendrik Lenstra made the following conjecture in private communication: Let $n \geq 2$. If a degree $n$ monic integer polynomial $P$ is chosen at random, then $\mathbb{Z}[x]/P(x)$ is integrally closed with probability $6/\pi^2$. So, in some sense, monogenic fields are not that rare. We talked about these questions before [here](https://mathoverflow.net/questions/167939/) and [here](https://mathoverflow.net/questions/176697/).


