Generating the Group of Nonzero Elements of a Quadratic Extension of Fp\mathbb{F}_{p}

It is well known that if $\mathbb{F}$ is a finite field then $\mathbb{F^{*}}$, the set of non zero elements of $\mathbb{F}$, is a cyclic group. In this paper we will assume $\mathbb{F}=\mathbb{F}_{p}$ (the finite field with p elements, p a prime) and $\mathbb{\mathbb{F}}_{p^{2}}$ is a quadratic extension of $\mathbb{F}_{p}$. In this case, the groups $\mathbb{F}_{p}^{*}$ and $\mathbb{F}_{p^{2}}^{*}$ have orders $p-1$ and $p^{2}-1$ respectively. We will provide necessary and sufficient conditions for an element $u\in\mathbb{F}_{p^{2}}^{*}$ to be a generator. Specifically, we will prove $u$ is a generator of $\mathbb{F}_{p^{2}}^{*}$ if and only if $N(u)$ generates $\mathbb{F}_{p}^{*}$ and $\frac{u^{2}}{N(u)}$ generates Ker$\,N$, where $N:\mathbb{F}_{p^{2}}^{*}\rightarrow\mathbb{F}_{p}^{*}$ denotes the norm map. We will also provide a method for determining if $u$ is not a generator of Ker$\,N$

Implementing electronic substance use disorder and depression and anxiety screening and behavioral interventions in primary care clinics serving people with HIV: Protocol for the Promoting Access to Care Engagement (PACE) trial.

BackgroundSubstance use disorders (SUDs) and psychiatric disorders are common among people with HIV (PWH) and lead to poor outcomes. Yet these conditions often go unrecognized and untreated in primary care.MethodsThe Promoting Access to Care Engagement (PACE) trial currently in process examines the impact of self-administered electronic screening for SUD risk, depression and anxiety in three large Kaiser Permanente Northern California primary care clinics serving over 5000 PWH. Screening uses validated measures (Tobacco, Alcohol, Prescription medication, and other Substance use [TAPS]; and the Adult Outcomes Questionnaire [AOQ], which includes the Patient Health Questionnaire [PHQ-9] and Generalized Anxiety Disorder [GAD-2]) delivered via three modalities (secure messaging, tablets in waiting rooms, and desktop computers in exam rooms). Results are integrated automatically into the electronic health record. Based on screening results and physician referrals, behavioral health specialists embedded in primary care initiate motivational interviewing- and cognitive behavioral therapy-based brief treatment and link patients to addiction and psychiatry clinics as needed. Analyses examine implementation (screening and treatment rates) and effectiveness (SUD, depression and anxiety symptoms; HIV viral control) outcomes using a stepped-wedge design, with a 12-month intervention phase implemented sequentially in the clinics, and a 24-month usual care period prior to implementation in each clinic functioning as sequential observational phases for comparison. We also evaluate screening and treatment costs and implementation barriers and facilitators.DiscussionThe study examines innovative, technology-facilitated strategies for improving assessment and treatment in primary care. Results may help to inform substance use, mental health, and HIV services.Trial registrationNCT03217058