The primary aim of the paper is to prove, 
within Bishop-style constructive mathematics and by 
elementary means (that is, without employing the full functional calculus), 
that under suitable conditions on the Banach algebra, 
its positive elements have positive square roots. 
This requires careful definition of such notions as positivity for Banach-algebra elements. 
The results are then applied to a problem about principal ideals in a Banach algebra.