So I am wondering if anyone recognizes the following theorem:
Given a prime $p$, and a base $b$ (natural number $>1$), the period of $\frac{1}{p}$ expressed in base $b$ is the unique $d$ that divides $p-1$ such that $b\mod p$ is a solution of $\Phi_d(x)\cong 0\mod p$ (where $\Phi_d(x)$ is the $d$th cyclotomic polynomial).
The reason I ask is that I proved this theorem while doing some original research a number of years back, and have been unable to find it anywhere in the literature (though this might be due to me not knowing much ring and field theory).
I am assuming that $\gcd(b,p)=1$.
It is easy to show that the length $n$ of the period $d$ of $1/p$ in base $b$ is the order of $b$ in the group $\Bbb{Z}_p^*$. So $n$ is the smallest positive integer such that $p\mid b^n-1$ In this form you can probably find it in textbooks, but I cannot point you at one. I have used this as an exercise though, and I certainly didn't come up with the result myself. I just don't remember how I first heard/read about it.
Now, by the basic properties of cyclotomic polynomials we have the factorization $$ x^n-1=\prod_{d\mid n}\Phi_d(x). $$
Your result follows from plugging $x=b$ into this. $\Phi_d(b)$ is a factor of $b^d-1$, so the smallest $n$ such that $p\mid b^n-1$ is also the smallest $n$ such that $p\mid\Phi_n(b)$.