Arithmetic properties of direct product of 𝑝-adic fields elements, II
https://doi.org/10.22405/2226-8383-2021-22-2-236-256
Abstract
statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔
∼=Ω𝑝1 ⊕. . .⊕Ω𝑝𝑛, a subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Here are
appropriate theorems for numbers not only like 𝛼 =∞Σ︀𝑗=0𝑎𝑗𝑔𝑟𝑗 where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded. But, for numbers 𝑓(𝛼), where 𝑓(𝑧) =∞Σ︀𝑗=0𝑐𝑗𝑧𝑗 ∈ Z𝑔[[𝑧]].
Furthermore, let ̂︀ Q∼=Π︀𝑝Q𝑝 be the ring of polyadic numbers, then, the article takes a look at̂︀ Ω=Π︀𝑝Ω𝑝, there are similar results for numbers like 𝑓(𝛼), where 𝑓(𝑧) =∞Σ︀𝑗=0𝑐𝑗𝑧𝑗 ∈ ̂︀ Z[[𝑧]],𝛼 =∞Σ︀𝑘=1
𝑎𝑘𝑔𝑟𝑘 , 𝑎𝑘 ∈ Z𝑔, 𝑔 = (𝑝1, . . . , 𝑝𝑛, . . .)
About the Author
Aleksei Sergeevich Samsonov
Moscow State Pedagogical University
Russian Federation
Russian Federation
graduate student
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Review
For citations:
Samsonov A.S. Arithmetic properties of direct product of 𝑝-adic fields elements, II. Chebyshevskii Sbornik. 2021;22(2):236-256. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-2-236-256
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