Multidimensional local skew-fields (792481), страница 22
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Bq2 = 0. Really,its discriminant must be equal to q12 p21 l2 , l ∈ Z. But D = q12 j 2 p21 + 4q12 p21 j(j − n1 ) +4q13 p1 (qa − 1)(3j + n1 ), hence 0 < p21 q12 (j − n1 )2 = q12 p1 (qa − 1)(3j + n1 ) < 0, acontradiction. Thus we have proved the case B) 2) b) ii) i’).If n1 |m1 or (m1 /n1 )k1,n1 = k2,m1 , then the solvability of the equation (1.26)for m = nq+1 , a = anq +m1 +i(α)−1 , b = cnq +m1 +i(α)−1 doesn’t depend on coefficientsanq +n1 +i(α)−1 (cnq +n1 +i(α)−1 ). In this case for almost all q fnq can be chosen so that theequation (1.26) is solvable for m = nq+1 , and fmq so that equation (1.26) is solvablefor m = mq+1 .
Not very complicated modification of all arguments, mentioned before,leads us to the conclusion, that α is conjugated to β:β(u) = ξu + Az j ,β(z) = z + Bz i(α) + B2 z i(α)+n1 + B3 z i(α)+m1 + B4 z i(α)+2m1 + Bq z i(α)+m1 + Bq2 z i(α)+nq2 ,where Bq = cuk2,mq −2+qa or equals to zero in accordance with equality to zero of theexpressionq1 (1 − qa − (q − 1)p1 ),B4 = Bq for q = 1, Bq2 = cuk1,nq2 −2+qa or equals to zero in accordance with equality tozero of the expression∂∂(b)/j − b (A)/(jA) + (nq2 − j)p1 u−1 b/(jq1 ) + a = 0∂u∂u103with b = nq2 −1 uk1,nq2 −1 +k2,m1 −2+qa , a = (k2,m1 − 2 + qa )uk1,nq2 −1 +k2,m1 −3+qa + (1 − i(α) +k+k−3+qa(p1 − qa + k1,nq2 −1 )/j, i.e.nq2 −2 )u 1,nq2 −1 2,m1−n21 p1 −q1 (p1 (2j+q1 (q2 −3)2 )−(j+q1 (q2 −2))(qa −1))+n1 q1 (−1−2p1 (q2 −3)+qa ) (1.36)This equation has no solutions in integers by the same reasons as (1.35), wherefrom Bq2 = 0 (see case B) 2) b) ii) ii’) ).In the case 2)m1 < n1 , and since j |q1 , we can apply here the arguments from the case 1).
Then theresult would coincide with the result of the previous paragraph.In the case 3)m1 < n1 , but j|q1 , so we rewrite the formula (1.29) for the conjugation fmq , q ≥ 1 inthe following way:α fmq (u) = ξu + Az j + cmq +m1 +j z mq +m1 +j + . . . + ξxmq z mq +mq Bxmq z mq +j +∂(xmq )Az mq +j +∂u1 ∂2∂2(xmq )A2 z mq +2j + CmB 2 xmq z mq +2j + mq B (xmq )Az mq +2j +q22 ∂u∂umq B2 xmq z mq +m1 +j + mq B3 xmq z mq +n1 +j + . . .∂(A)xmq z mq +j +∂u+ . . . + cmq +m1 +j z mq +m1 +j + . .
.fmq α (u) = ξu + ξxmq z mq + Az mq + Az j +jymq Az mq +j + cnq +j z nq +jα fmq (z) = z+Bz i(α) +B2 z i(α)+m1 +B3 z i(α)+n1 +amq +m1 +i(α)−1 z mq +m1 +i(α) +. . .+ymq z mq +1 +2∂mq +i(α)mq +i(α) 1 ∂2+(mq +1)Bymq z+(ymq )A2 z mq +2i(α)−1 +CmB 2 ymq z mq +2i(α)−1 +(ymq )Azq +12∂u2 ∂u∂(mq + 1)B (ymq )Az mq +2i(α)−1 + . . . + (mq + 1)B − 2ymq z mq +m1 +i(α) + .
. .∂ufmq α (z) = z + ymq z mq +1 + Bz i(α) + B2 z i(α)+m1 + B3 z i(α)+n1 +∂∂(B)xmq z mq +i(α) + i(α)ymq Bz mq +i(α) +(B2 )xmq z i(α)+m1 +mq +∂u∂uanq +i(α)−1 z nq +i(α) + . . . + amq +m1 +i(α)−1 z mq +m1 +i(α) + (i(α) + m1 )ymq B2 z m1 +mq +i(α) + . . .Hence follows, that two cases are possible:a’) the solvability of the equation (1.26) for m = mq + m1 doesn’t depend oncoefficients cnq +j , .
. . , cmq +m1 , anq +j , . . . , an1 +m1 , that is it depends only on cmq +m1 +j ,amq +m1 +i(α)−1 (and it is equal to −p1 + qa − 2 = q1 (qa − 1)/j).104b’) the solvability of the equation (1.26) for m = mq + m1 depends on coefficientscnq +j , . . . , cmq +m1 , anq +j , . . . , an1 +m1 , i.e.
−p1 + qa − 2 = q1 (qa − 1)/j.In the case a’), repeating the proof as in the previous cases, we get the same resultas in the case 2) (it corresponds to a case B) 2) c) ii) i’)).In the case b’) the conjugations fmq determine the solvability of the equation (1.26)for m = nq , and conjugations fnq — for m = mq+1 . Here, in this case of m = nq , as seenfrom the formula above, can exist not more than two q — solutions of the equation∂∂(b)/j − b (A)/(jA) + (nq2 − j)p1 u−1 b/(jq1 ) + a = 0∂u∂u2∂∂2qa2+ Cmu2qa −2 p21 /q12 uk2,mq + mq p1 /q1 u2qa −1 ∂u(uk2,mq ),with b = 2−1 ∂u2 (uk2,mq )uq∂22qa2(p1 −qa +k2,mq )/j+Cmu2qa −2 p21 /q12 uk2,mq −1 (p1 −qa +k2,mq )/j+a = 2−1 ∂u2 (uk2,mq −1 )uq +1∂(uk2,mq −1 )(p1 − qa + k2,mq )/j, i.e.(mq + 1)p1 /q1 u2qa −1 ∂u(p21 (−1+q−q1 (−1+q−2q 2 +qq1 ))−qq12 (qa −1)+p1 q1 (1−3q−(q−1)qq1 +qqa )) = 0 (1.37)and in the case m = mq+1 not more than (mq+1 − nq )/j + 1 = q1 /j = w q — solutionsof the appropriate equation∂∂(b)/j − b (A)/(jA) + (nq2 − j)p1 u−1 b/(jq1 ) + a = 0∂u∂u(1.38)Thus, α is conjugated to β:β(u) = ξu + Az j ,β(z) = z + Bz i(α) + B2 z i(α)+m1 + Bqn ,1 z i(α)+nq1 + Bqn ,2 z i(α)+nq2 + Bqm ,1 z i(α)+mq1 + .
. . +Bqm ,w z i(α)+mqw ,k−2+qawhere Bqn ,i = ci uk1,nqi −2+qa , Bqm ,j = cj u 2,mqjor 0 depending on solvability ofcorresponding equations (it is the case B) 2) c) ii) ii’)).In the case c),when q1 = j, we use arguments from both: the previous case and the case a), and getfrom there, that α is conjugated to β:β(u) = ξu + Az j ,β(z) = z + Bz i(α) + B2 z 2i(α)−1 + B3 z 3i(α)−2 + Bqm,1 ,1 z i(α)+q0 q1 + Bqm,1 ,2 z i(α)+q2 q1 +Bqm,1 ,3 z i(α)+q3 q1 + Bqm,2 ,1 z i(α)+q1 q1 + Bqm,2 ,2 z i(α)+q2 q1 + Bqm,2 ,3 z i(α)+q3 q1 ,−1where B2 = cb2 u−p1 −1+qa , B3 = cb3 u−p1 +2qa −2 , Bqm,1 ,i = cbqm ,1 ,i u−p1 q1 (qi q1 −j)+2qa −2 ,Bqm,2 ,j = cbqm ,2 ,j u−p1 qj +qa −1 or 0 depending on solvability of corresponding equations∂∂(b)/j − b ∂u(A)/(jA) + (nq2 − j)p1 u−1 b/(jq1 ) + a = 0. If we denote as b50 , a50 b and∂ua in (1.37), then the appropriate b, a, b , a for two equations are equal tokmq ,1 −p1 +1−2+qaub = b50 + c−2a mq u105−p1 +qa −2+qa kmq ,1 −1a = a50 + c−2u(p1 − qa + k1,mq )/ja (1 − i(α) + mq − m1 )u(1.39)kmq ,2 −p1 +1−2+qaub = b50 + c−2a mq u−p1 +qa −2+qa kmq ,2 −1a = a50 + c−2u(p1 − qa + k2,mq )/ja (1 − i(α) + mq − m1 )u(1.40)where ca is a constant for the coefficient A, and the rest notations are taken from thecase a).
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