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A Boundary Value Problem for equations whose order by Zainulabidova Z.M.

By Zainulabidova Z.M.

Translated from Differentsial'nye Uravneniya, Vol. forty, No. 10, 2004.

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Let p ∈ Z be a prime, let k > 0 be an integer, and let τ be a nonzero prime ideal of E such that p ∈ τ k ⊂ E(p) ⊂ E, and such that E(p)/τ k is a field. 5. Let [E(p)/τ k : Z/pZ] = e ≥ 1 and [E/τ : E(p)/τ k ] = f ≥ 1. Then [E/τ : Z/pZ] = ef ≥ 1, and Z/pZ-dim(τ/τ k ) = ef (k − 1) ≥ 0. 7. 6) occurs in the following natural way. Let p be a rational prime, let p ∈ τ ⊂ E be a prime ideal, and let k ≥ 1 an integer such that p ∈ τ k . Then E/τ is a field and E/τ k is a finite Z/pZ-algebra with Jacobson radical τ/τ k .

8, (u(E) + τ )/τ = 1. Since p > 2, u(E/τ ) = 1, so some element of u(E/τ ) does not lift to an element of u(E). This completes the proof. 10. Let E be an rtffr Dedekind domain with finite unit group. There are at most finitely many prime ideals τ ⊂ E and finitely many integers k > 0 such that τ k ∩ Z is a prime ideal in Z, and such that each unit of E/τ k lifts to a unit of E. Proof: Let card(u(E)) = m be finite. The set P = {primes p ∈ Z there are integers s > 0 and t ≥ 0 such that pst (ps − 1) ≤ m} is finite.

2 Analytic methods We link the power cancellation property to the sequence of rational primes. 1 as follows. Let k be an algebraic number field with E its ring of algebraic integers and with class number h(k). Let G be an rtffr abelian group with class number h(G). 14. 1. Let k be an algebraic number field, let [k : Q] = f , and let h(k) denote the class number of k. Let E be the ring of algebraic integers in k. Then E is a ring whose additive group (E, +) is a free abelian group of finite rank f .

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