Prime numbers in certain arithmetic progressions (Q873781)

From the text: The authors discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). The first author [J. Madras Univ., Sect. B 51, 161--169 (1988; Zbl 1194.11093)] had shown earlier that such proofs can exist if and only if the residue class (mod \(k\)) has order 1 or 2. After reviewing this work (as it is difficult to obtain), they consider generalizations of this question to algebraic number fields. The main tool will be the Chebotarev density theorem showing that the class of polynomials needed for a Euclidean proof does not exist unless \(\ell^2\equiv 1\pmod k\).