According to Dickson, [1, p. 181], it was d'Alembert in 1769 who claimed the
existence of a primitive root modulo a prime p. He didn't attempt to prove
this fact.
Euler remarked in [2, \S 37] that this fact needs a proof, as there is no
primitive root for most composite numbers. His attempt [2, \S 38] to prove it
failed as Gauss pointed out [Disq. art. 56]. Euler's failure is not very
serious. He needed that the congruence
x^n - 1 = 0 modulo p
has at most n solutions modulo p, if p is a prime. If he had claimed that any
algebraic equation of degree n has at most n solutions modulo p, his proof
(eine Puenktcheninduktion, as we call the type of proof he uses in German) would
have been perfect [2, \S\S 37, 28]. Being more general pays quite often! This
more general theorem was proved by Lagrange (Me/m. de l'Ac. de Berlin,
Anne/e 1768, p. 192. Quoted after Gauss, Disq. art 44.). So Euler could have
referred to Lagrange.
Gauss gave two proofs for the existence of a primitive element [Disq., artt.
52, 53]. The first one is most beautiful and yields much more than stated by
Gauss.
[1] L. E. Dickson, History of the Theory of Numbers, Vol. 1. Reprint of the
edition 1919. New York 1971
[2] L. Euler, Demonstrationes circa residua ex divisione potestatum per numeros
primos resultantia. Novi commentarii academiae scientiarum Petropolitanae 18
(1773), 85-135, 1774, Opera omnia II, 240-281
Heinz Lueneburg
> ***********
> do we find, and where, in Euler 1) the notion of "minimum power"
> (in French: "exposant minimal") and 2) the possibility of expressing
> any residue by a power of a primitive root?
> **********
>
> Udai Venedem
> venedem@wanadoo.fr
> http://perso.wanadoo.fr/alta.mathematica/