On Existence of Euclidean Ideal Classes in Real Cubic and Quadratic Fields with Cyclic Class Group

On the existence of euclidean ideal class in quadratic, cubic and quartic extensions

“…If p, q, r are the odd primes, then there exists a Euclidean ideal class in Cl K . [6], also examined the existence of a Euclidean ideal in the case of abelian number fields K with a prime class number, rank(O × K ) ≤ 2. They proved the following two theorems.…”

“…Theorem 6 (Gun and Sivaraman [6]). Let K 1 , K 2 , K 3 be distinct real quadratic fields with prime class number having conductors f 1 , f 2 , f 3 respectively.…”

“…Theorem 12 (Gun and Sivaraman [6]). Suppose that K is a number field with rank(O × K ) ≥ 1 and Cl K = [a] .…”

“…Proof. Refer to [6,Lemma 16 ]. Now we end this section by stating a famous conjecture in Sieve Theory.…”

“…As K is real cubic field, O × K contains two multiplicative independent elements, say e 1 , e 2 . Assuming Elliott and Halberstam Conjecture is true, by [6,Lemma 19], there exists a set P ⊆ M ǫ such that for every p ∈ P for some η ∈ {±e 1 , ±e 2 } is primitive modulo p and there exists an increasing sequence {x n } n∈N such that |P (x n )| ≫…”