By Zhukavets N.M., Shestakov I.P.

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H2 ) does not capitulate in K5 (resp. p K6 ). x C y d/ be the fundamental unit of Q. y1 1 4 C p y2 /: The class of the ideal H H comes principal in K if and only if 2 3 1 2 5 the equation 1 2 " D ˛ 2 has a solution in K5 , where ˛ 2 K5 and " is a unit p of K5 . Looking for a solution ˛ 2 K5 of the form ˛ D ˛2 1 3 , where ˛2 2 k. p 2 " D ˛22 3 , this show that " is a unit of k. ˛2 1 3 / ; hence . e. ˛/, so H1 H2 capitulate in K5 . d/ (iii) Using examples, we show that in the extension K5 , H0 2 capitulate in some cases and does not in others.

1. 1 W n/ is Z-graded, deg t D 1, deg @@ i D 0 and satisfies all the conditions above. t; deg @t@ D Suppose that n 2. Let H be the F-span of all elements i @@ i j Ä n. 2/-triple @ @ j@ j; j@ i. 0/ is the centralizer of H in L. FŒt ; t C FŒt ; t 1 D 3 Graded Modules over Superconformal Algebras 43 P 2. D/ D . 1/jfi j @@fii C @f@t . Let ˛ 2 F. n; ˛/ is simple unless ˛ 2 Z. n; ˛/ C Ft ˛ 1 n @t . n; ˛/ is simple. n; ˇ/ if and only if ˛ ˇ 2 Z. @ Let n 2. As above, H D span. i @@ i j @ j /, 1 Ä i ¤ j Ä n, the elements ˛CkC1 k @ ek D tkC1 @t@ t .

P1 p2 ; i/, where p1 and p2 are primes such p1 that p1 Á 1 mod 8, p2 Á 5 mod 8 and . p2 / D 1. Lecture Notes in Pure and Applied Mathematics, vol. 208. Dekker, New York, pp 13–19p (2000) 2. A. Azizi, Sur le 2-groupe de classe d’idéaux de Q. d; i/. Rend. Circ. Mat. Palmero (2) 48, 71–92 (1999) 3. A. Azizi, Unités de certains corps de nombres imaginaires et abéliens sur Q. Ann. Sci. Math. Que. 23(1), 15–21 (1999) 4. A. Azizi, Construction de la tour des 2-corps de classes de Hilbert de certains corps biquadratiques.