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algebra of inertia algebraic extension algebraic functions arbitrary central simple algebra chapter coefficients complete residue system congruences 8.22 conjugate pairs consequently consider convenient form corresponds crossed product cyclic pair defined division algebra equal exists factor set field of algebraic field of inertia follows form f formal power series h mod Hence I. M. Vinogradov inequality integer integral quaternion invariants L. E. Dickson L. K. Hua large numbers Lemma line of formula Linnik Math mod q mod4 mod8 modp modr modulo q number of distinct number of solutions numbers as sums obtain polynomial predicate prime divisor primitively representable proper quaternion proper vector proves the lemma quantifiers quaternion equation quaternion of norm reduced residue system relatively prime representations of numbers residue system modulo satisfy the congruence simple central algebra singular series solvable sufficiently large sum of three Suppose values variables Waring's problem