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《Mathematical Reviews》Selected Matches for: Items Authored by Zhang, Xian Ke (45 Reviews,partial) |
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| 美国《数学评论》对张45篇论文的评论(部分) |
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A real quadratic field $k=\bold Q(\sqrt D)$ is said to be of Richard-Degert (RD) type if its discriminant can be written in the form $D=m\sp 2+r$ with $r\mid 4m$. The authors study abelian octic fields $K$ with Galois group isomorphic to $C\sb 2\times C\sb 2\times C\sb 2$, where $C\sb 2$ denotes a cyclic group of order 2 containing three quadratic subfields $k\sb 1, k\sb 2, k\sb 3$ of RD type. They show that, under certain conditions on the discriminants of the fields $k\sb i$, $i=1,2,3$, the ideal class number of $K$ is equal to the product of the class numbers of seven quadratic subfields of $K$ divided by an explicit power of 2. They also show that the fundamental units of $K$ can be expressed in a fully explicit way in terms of the fundamental units of the quadratic subfields of $K$. These results generalize previous results of G. Frei [Arch. Math. (Basel) 36 (1981), no. 2, 137--144; MR 82i:12005] for biquadratic fields consisting of quadratic fields of RD type.
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Let $K$ be a number field, let ${\scr O}\sb K$ be the ring of integers of $K$ and let $M$ be a finitely generated module over ${\scr O}\sb K$. Then $M=M\sb{\rm tors}\oplus {\scr O}\sb K\sp m\oplus {\scr A}$, where $\scr A$ is an ideal of $K$. The class $[\scr A]$ is uniquely determined; $[\scr A]$ is called the Steinitz class of $M$ and is denoted by ${\rm St}(M)$. The authors prove that $\scr A$ can be chosen such that $N\sb{Q}\sp K{(\scr A)}=(L\colon\!M)$ (provided $L$ is a free ${\scr O}\sb K$-module, $M\subset L$ and $(L\colon\! M)<+\infty$). They apply this result to determining the Steinitz class of the Mordell-Weil group of some elliptic curves. To be more precise, let $K={Q}(\sqrt{-p})$ ($p$ a prime) be an imaginary quadratic field and let $E$ be an elliptic curve having complex multiplication by ${\scr O}\sb K$. Let $F={Q}(j(E))$ and $H=K(j(E))$. Then $E$ is defined over $F$, $H$ is the Hilbert class field of $K$, $(H\colon\!F)=2$, and the Mordell-Weil group $E(H)$ has a natural structure of a module over ${\scr O}\sb K$. If $p\equiv 3\bmod 4$, the authors put $L={\scr O}\sb K·E(F)\sb f$, and $M=[\sqrt{-p}]E(H)\sb f$ (where the index $f$ denotes the free part of a module and $[\alpha]$ denotes the endomorphism of $E$ corresponding to $\alpha \in {\scr O}\sb K$). In this case they prove that ${\rm St}(E(H))$ (or equivalently ${\rm St}(M)$) is the trivial class. If $p\equiv 1\bmod 4$, the authors put $L=$ ${\scr O}\sb K·E(F)\sb f$ and $M=[2\sqrt{-p}]E(H)\sb f$ and prove that ${\rm St}(E(H))=[{\scr P}]\sp t$, where $\scr P$ is a prime factor of 2 in $K$, $t=l+\log\sb 2\vert H\sp 1(G,E(H)\sb f)\vert $, $l={\rm rank}\sb {Z}E(F)$ and $G={\rm Gal}(H/F)$. They also demonstrate their method in the case of elliptic curves from [D. S. Dummit and W. L. Miller, J. Number Theory 56 (1996), no. 1, 52--78; MR 96k:11066]. Finally, they state the following conjecture: Both the cases ${\rm St}(E)=1$ and ${\rm St}(E)\neq 1$ exist for some elliptic curve $E$ having complex multiplication by ${\scr O}\sb K$, where $K={Q}(\sqrt{-D})$ with prime number $D\equiv 1\pmod 4$.
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Summary: "For elliptic curves $E$ over the rationals $\bold Q$, the classification according to their torsion subgroups $E\sb {\rm tors}(\bold Q)$ of rational points is studied. When $E\sb {\rm tors}(\bold Q)$ are cyclic groups of even orders, the classification is given with explicit criteria, and the generators of the torsion groups are also explicitly presented in each case. These results, together with the recent results of Ono for the non-cyclic torsion groups, completely solve the problem of the explicit classification with $E$ having a rational point of order 2."
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The author presents explicit formulas for the discriminant and factorization of primes in abelian extensions of the rationals whose Galois group is a $p$-group.
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This paper is a continuation of a paper by the authors [Kexue Tongbao (Chinese) 42 (1997), no. 19, 2053--2056; MR 99g:11128a]. The problem with both these papers is that the crucial notion of potential element of order $p$ in the class group is too loose to make it possible to prove any of the conjectures put forward by the authors. We suspect that the correct definition should be as follows. Let $p\ge2$ and $q\ge2$ be two given primes. The ideal class group of the real quadratic field $K$ has a potential element of order $p$ above $q$ if (i) $(q)=QQ'$ splits in $K$ and (ii) the ideal $Q\sp p$ is principal. For example, for $n\sp 2-27$ positive and square-free, the $K\sb n={Q}(\sqrt{n\sp 2-27})\text{'s}$ have a potential element of order 3 above 3, for $27=N\sb {K\sb n/{Q}}(n+\sqrt{n\sp 2-27})$. The idea is that if $p$ and $q$ are given and if we let $K$ range over a family of real quadratic fields for which their ideal class groups have a potential element of order $p$ above $q$ then most often $p$ should divide the class number of $K$, for the ideal class of $Q$ should be of order $p$. The authors suggest modifications of the Cohen-Lenstra heuristics [see H. Cohen and H. W. Lenstra, Jr., in Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), 33--62, Lecture Notes in Math., 1068, Springer, Berlin, 1984; MR 85j:11144] to predict the probability of both these events.
Reviewed by Stephane Louboutin
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99g:11128b
11R11 (11R29 11Y40)Consider, for example, the family of real quadratic fields $K\sb n=\bold Q(\sqrt{n\sp 2-27})$, $n\geq 4$. The class number $h\sb n$ of $K\sb n$ is often a multiple of 3 and there is a simple explanation for this, namely, $27=N\sb {K\sb n/\bold Q}(n+\sqrt{n\sp 2-27})$, so the cubes of the prime ideals of $K\sb n$ above 3 are principal. The authors suggest modification of the Cohen-Lenstra heuristics [see H. Cohen and H. W. Lenstra, Jr., in Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), 33--62, Lecture Notes in Math., 1068, Springer, Berlin, 1984; MR 85j:11144] to predict the probability that 3 divides $h\sb m$ and the probability that the prime ideals of $K\sb n$ above 3 are not principal.
Reviewed by Stephane Louboutin
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Summary: "A necessary and sufficient condition is given for the ideal class group $H(m)$ of a real quadratic field ${Q}(\sqrt{m})$ to contain a cyclic subgroup of order $n$. Some criteria satisfying the condition are also obtained. Eight types of such fields are proved to have this property, e.g. fields with $m=(z\sp n+t-1)\sp 2+4t$ (with $t\mid z\sp n-1)$, among which are the well-known fields with $m=4z\sp n+1$ and $m=z\sp {2n}+4$ as special cases."
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The author derives some elementary facts about discriminants of cyclotomic fields $K$ and the prime decomposition in $K/{Q}$ in terms of the character group associated to $K$.
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Sufficient conditions are given for an abelian extension $L/K$ to have a relative integral basis when ${\rm Gal}(L/\bold Q)\cong (\bold Z/q\sp s \bold Z)\sp n$ and ${\rm Gal}(K/\bold Q)\cong (\bold Z/q\sp r\bold Z)\sp m$. Here $L=K\sb 1K\sb 2\cdots K\sb n$, where each $K\sb i/Q$ is a cyclic extension of degree $q\sp s$. For any prime $p$, and an abelian extension $F/\bold Q$, let $e(p,F)$ denote the ramification index of $p$ in $F$ and $q\sp {e\sb p}=\max\sb {1\leq i\leq n} e(p,K\sb i)$.
Let $[L\colon\!K]=q\sp M$. If $q$ is an odd prime and $M\geq e\sb p$ for all prime numbers $p$ with $e(p,L)\neq e(p,K)$ then $L/K$ has a relative integral basis. For $q=2$, a similar but more complicated condition is given for the existence of a relative integral basis.
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Let $O\sb L$ and $O\sb K$ be the rings of the algebraic integers of the number field $L$ and its subfield $K$, respectively. We say that $L/K$ has a relative integral basis if $O\sb L$ is a free $O\sb K$-module. In this paper, the existence of the relative integral basis of abelian $q$-fields is discussed and the main results are as follows. Theorem A. Let $L\supset K$ be an abelian $q$-field with $q\not=2$. If $[L\colon K]\geq e(p,L)$ holds for every prime $p$ such that $e(p,L)\not=e(p,K)$, then $L/K$ has a relative integral basis, where $e(p,\Omega)$ denotes the ramification index of $p$ in the field $\Omega$. Theorem B. Let $L\supset K$ be an abelian 2-field. If $L/K$ is not a cyclic extension and $[L\colon K]>e(p,L)$ if $p\not=2$ or $L/K\in C\sb 2(1,1),\ [L\colon K]>e(p,K)$ if $L/K\in C\sb 2(2,1)$ holds for every prime $p$ such that $e(p,L)\not=e(p,K)$, then $L/K$ has a relative integral basis. The paper is one of a series of the author's papers on the same topic and the proof is based on E. Artin's ideas.
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Let $L/\bold Q$ be an abelian extension of the rationals whose degree is a prime power and let $K$ be a subfield of $L$. The author sketches the proof of a sufficient condition for the extension $L/K$ to have a relative integral basis.
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In this paper the author announces some recent results on the Diophantine equation $x\sp 2-dy\sp 2=c$. Details of the proofs will appear elsewhere.
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In this paper the author aims to study the Diophantine equation $x\sp 2-dy\sp 2=c$ for a positive square-free integer $d$ and to obtain structure theorems on ideal class groups of real quadratic fields. Namely, via the theory of semisimple (generalized) continued fractions he provides simple criteria for solvability and explicit sets of solutions of the Diophantine equation, without proofs. He uses these results in order to obtain some structure theorems and to improve a theorem of Cohen-Lenstra under certain conditions.
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If $L$ is a Galois extension of $\bold Q$ with Galois group $(\bold Z/q\sp s\bold Z)\sp n$ for some odd prime $q$ and $K$ is a subfield of $L$ with ${\rm Gal}(K/\bold Q)=(\bold Z/q\sp s\bold Z)\sp m$ for $m\leq n$ then it is shown that $L/K$ has a relative integral basis. If $q=2$ and $n-m>1$ then the result also holds. In the latter case some additional hypothesis is necessary [e.g. see R. H. Bird and C. J. Parry, Pacific J. Math. 66 (1976), no. 1, 29--36; MR 55 #5579].
The proof of the results given here depends on showing that the Steinitz class of the extension $L/K$ is principal. It has been shown by E. Artin [in Algebre et theorie des nombres, 19--20, CNRS, Paris, 1950; MR 13, 113i] that $(D/\Delta)\sp {1/2}$ is in the Steinitz class. Here $D=D(L/K)$ is the relative discriminant of $L/K$ and $\Delta=\Delta(L/K)$ is the discriminant of any $K$-basis of $L$. The author determines $D$ and shows that it is a rational square. To complete the proof an argument is given showing that $\Delta$ is a square in $K$.
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Summary: "All integer solutions and all $c$'s for which there is a solution are determined by formulae for general Diophantine equations $x\sp 2-my\sp 2=c$. The determination is most explicit for several types of $m$. For example, if $m=n\sp 2+1$ then $\vert c\vert =1$, $2n$, $4n±3$, $6n±8$, $8n±15$, $10n±24$, or $\vert c\vert \geq 12n-35$ and $6n+1$. The corollaries greatly develop corresponding results for small $c$ due to Ankeny, Hasse, S.-D. Lang, Takeuchi, Yokoi and Mollin, and lead to a series of results on the class numbers $h(m)$ of real quadratic fields $\bold Q(\sqrt m)$; e.g., $n$ divides $h((4\sp \delta z\sp n+t-1)\sp 2+4t)$ with $t\mid 4\sp \delta z\sp n-1$, $\delta=0,1$."
\{For the entire collection see MR 95e:00029\}.
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For a real quadratic field $K\colon\bold Q(\sqrt m)$ ($m$ square-free), if $m$ is expressed in the form $m=s\sp 2+r$, $r\mid 4s$, then $K$ is said to be of extended Richaud-Degert type (simply ERD type). Similar to the case of $r\mid 4s$ in ERD type, the author studies the ideal class group of $K$ in both the cases $r\mid 6s$ and $r\mid 8s$, and provides without proof some results on the order of the ideal class containing, respectively, the ideal $(3,s+\sqrt m)$ in the case $r\mid 6s$ and the ideal $(2,(1+\sqrt m)/2)$ in the case $r\mid 8s$.
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Let $K$ be a cyclic quartic number field, and $k$ its quadratic subfield. Let $h(L)$ denote the ideal class number of field $L$. Ten congruences for $h\sp -=h(K)/h(k)$ are obtained. In particular, if $K=\bold Q(\sqrt{p+s\sqrt p})$ with the prime number $p=r\sp 2+s\sp 2$ and $s$ is even, then $C\sb 1h\sp -\equiv B\sb {(p-1)/4}B\sb {3(p-1)/4}\bmod p$ for $p\equiv 1\bmod 8$; and $C\sb 2h\sp -\equiv E\sb {(p-5)/8}E \sb {(3p-7)/8}\bmod p$ for $p\equiv 5\bmod 8$, where $B\sb n$ and $E\sb n$ are the Bernoulli and the Euler numbers.
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Let $m>3$ be a square-free integer. The author considers the class numbers $h(m)$ and $h(-m)$ of the real quadratic field $\bold Q(\sqrt m)$ and imaginary quadratic field $\bold Q(\sqrt {-m})$. He shows that there exist simple congruences mod $8$ between $h(-m)$ and the product of $h(m)$ by an explicit expression involving the coefficients $a$ and $b$ of the fundamental unit $\epsilon=(a+b\sqrt m)/2$ of $\bold Q (\sqrt m)$.
To prove this result the author uses essentially the 2-adic $L$-functions and a result due to A. Kudo [J. Math. Soc. Japan 27 (1975), 150--159; MR 50 #12968] that establishes the congruence $L\sb 2(1,\chi)\equiv L\sb 2(0,\chi)\bmod 8$ for certain Dirichlet characters $\chi$. We might note that, except for some slight differences, the results obtained here may be found in Kudo's article.
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Let $k=\bold F\sb q(t)$. The author proves that (1) all fields $k(\sqrt{D} ,\cdots,\sqrt{D\sb n})$ $(D\sb i \in\bold F\sb q[t], n\ge 2)$ with divisor-class-number one are as follows: $k(\sqrt{P\sb 1}, \sqrt{P\sb 2})$ and $k(\sqrt{P\sb 1P\sb 2}, \sqrt{P\sb 1P\sb 3})$, where the $P\sb i\in \bold F\sb q[t]$ are linear and distinct; (2) all imaginary fields $k(\sqrt{D\sb 1},\sqrt{D\sb 2})$ with ideal-class-number one are as follows: $D\sb 1=t$, $D\sb 2=t\sp 3-t-1 (q=3)$, $t\sp 2-t-1\ (q=3)$, $t\sp 2+2$ $(q=5)$ or $t+c (c\in\bold F\sb q)$ up to transformation $t\mapsto at+b$ $(a,b\in\bold F\sb q$, $a\ne0)$. The proof is based on the classification of fields of this kind he gave earlier [same journal Ser. A 31 (1988), no. 5, 521--530; see the preceding review] and on genus theory.
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This paper may be considered as a sequel to E. Artin's famous thesis treating quadratic extensions of the rational function field $k\coloneq\bold F \sb q(t)$ over the finite field $\bold F\sb q$, $q$ being a power of an odd prime. The author considers abelian extensions $L$ of $k$ having Galois group $G(L/k)$ isomorphic to $(\bold Z/2\bold Z)\sp n$. Then $L=k(\sqrt{D\sb 1},\cdots, \sqrt{D\sb n})$ where $D\sb i\in\bold F\sb q[t]$ are squarefree polynomials for $i=1, \cdots,n$. Consider such a field $L$ and let $P$ be a prime divisor in $k$. Let $P=(\germ P\sb 1\cdots\germ P\sb r)\sp e$ be the decomposition of $P$ in $L$ and let $f=[L\:k]/er$ be the residue degree of $\germ P\sb i$ over $P$. In Theorem 1 the author shows that for the triple $(e,f,r)$ there are only four possibilities. He gives an integral basis of $L/k$ (with respect to $\bold F\sb q[t]$, calculates the genus of $L$, the zeta-function of $L$, the (degree zero) class number of $L$ and the "conductor" of $L/k$---which can be defined according to a result of D. R. Hayes [Trans. Amer. Math. Soc. 189 (1974), 77--91; MR 48 #8444].
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The author considers ambiguous ideals, ambiguous ideal classes and genera in quadratic function fields $K=k(\sqrt{D(t)})$, where $k=\bold F\sb q(t)$. He derives a theory which is completely analogous to that for quadratic number fields and completes previous results of E. Artin [Math. Z. 19 (1924), 153--246; Jbuch 50, 107].
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For cyclic quartic fields $K$ the author states congruences for the class number, relative to the class number of a quadratic subfield, in terms of the conductor, the regulator, Bernoulli and generalized Bernoulli numbers, and a certain number $E$ of norm $\mp1$, which together with its Galois conjugate, $-1$ and the fundamental unit of the quadratic subfield generate the unit group of $K$.
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Summary: "Let $K$ be any cyclic cubic extension of the rationals $\bold Q$. Let $h,f,\langle\chi\rangle$ be its ideal class number, conductor, and character group. There exists $E=(x+y\tau+y\overline\tau)/3$, $x\in\bold Z$, $y\in\bold Q(\sqrt{-3})$, such that $E$ and its conjugate and $-1$ generate the unit group of $K$, where $\tau=\sum\chi(a)\exp(2\pi i/f)\sp a$ is a Gauss sum. Let $\omega$ be the Teichmuller character modulo $p$, $\chi\sb n=\chi \omega\sp {-n}$, $e=(p-1)/3$. We show that $hC\equiv\frac34B\sb {e,x\sb e}B\sb {2e,x\sp 2\sb e} \bmod p$ where $p\in\bold Z$ is any prime, $3\neq p\mid f$, and the constant $C=(x\sp 3-27)/(fx\sp 3)-y\overline y/x\sp 2$. In particular, if $f=p$ is a prime, then $hC\equiv\frac34 B\sb eB\sb {2e}\bmod p$, where $B\sb e\;[B\sb {e,x}]$ denotes a [generalized] Bernoulli number. For $p\nmid f$, a similar congruence is obtained."
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Let $l$ be a prime and let $K$ be an abelian extension of $\bold Q$ with Galois group $G\cong C\sb l\sp n$, where $C\sb l$ is the cyclic group of order $l$. The author proves, by studying the discriminant and the different, that for any subextension $k$ of $K$ with $[K\:k]>2$ there exists a relative integral basis for $K/k$. For the case of $[K\:k]=2$, the author finds out a relationship of the existence of relative integral basis and the existence of a unit in $k$ with certain properties. The last result has some consequences for units in $k$ which generalize those of H. Wada [J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 201--209; MR 35 #5414] and others.
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M. Bhaskaran [same journal 11 (1979), no. 4, 488--497; MR 80j:12002] gave a construction of the genus field of an arbitrary algebraic number field. The main oversight made in Bhaskaran's paper was for the case $p=2$, where it was incorrectly assumed that an abelian field $k$ of degree $e\sp *\sb p$ (over the rationals) and conductor $p\sp {\rho\sb p}$ is cyclic. It is for this case that the present author, in the paper under review, gives a counterexample, namely the field $k=\bold{Q}(\sqrt{2},\sqrt{3})$; and notes that there are many such counterexamples for the $p=2$ case. Moreover a remark is made which refers to Masuda's review of Bhaskaran's paper wherein there are reviewer's remarks including an example to show that the results are "rather odd". The author then gives a short proof of Bhaskaran's (corrected) result using the tools of class field theory.
Bhaskaran published a corrigenda [ibid. 19 (1984), no. 3, 449--451; MR 86i:11065], but he missed correcting case (ii) of the proof of Lemma 1 of his original paper. He did however correct the other fault with his paper, namely that the proof of Theorem 1 contained an incomplete induction especially when $p$ is odd. Several typos were also corrected.
It should also be noted that Bhaskaran published a later article [ibid. 21 (1985), no. 3, 256--259; MR 87e:12010] wherein Theorem 2 suffers from a similar oversight, namely that in the $p=2$ case it is incorrectly assumed that there is a unique abelian extension of the rationals of degree $e\sp *\sb p$ and conductor $p\sp {\rho\sb p}$. This is false for $e\sp *\sb 2=2$ since $\bold{Q}(\sqrt{-2})$ and $\bold{Q}(\sqrt{2})$ are choices, when the conductor is 8.
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Let $K$ be a quartic cyclic number field, $k={Q}(\sqrt{u})$ its unique real quadratic subfield, where $u=p\sb 1\cdots p\sb g$ and the $p\sb i$ are distinct primes. Then $K$ can be expressed in the form $K={Q}(\sqrt{\theta})$, $\theta=wv(cu+a\sqrt{u})$, where $c\sp 2u=a\sp 2+b\sp 2$, $a\equiv 1\bmod 4$, $(a,b)=(u,v)=1$, $0<w\vert u$, and $u$ and $v$ are square-free integers. Suppose $\varepsilon\sb 0=(s+t\sqrt{u})/2$ is the fundamental unit of $k$ with ${\rm N}\sb {k/{Q}}(\varepsilon\sb 0)=-1$. In the present paper the author proves that for fixed $u$ and $v$ there are exactly $2\sp {g-1}$ such quartic cyclic number fields $K$ and exactly one among them has relative integral basis over $k$. This is the field $K\sb 0={Q}(2\sp u v(tu+2\sqrt{u})\sp {1/2})$ and $\{1,(2\sp \delta+\sqrt{d\varepsilon\sp 3\sb 0})/2\}$ is a relative integral basis of it, where $d=2\sp \delta v\sb 1\sqrt{u}$ is the discriminant of $K\sb 0/k$ and $v\sb 1$ is the odd integer among $v$ and $v/2$.
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Integral bases for Galois extensions of degree four over the field of rationals were explicitly determined by A. A. Albert [Ann. of Math. (2) 31 (1930), 381--418; Jbuch 56, 870]. In the note under review, the author elaborates on claims made earlier [J. Number Theory 18 (1984), no. 3, 350--355; MR 85k:11053] that some of Albert's results are false, and corrected versions of these are given. \{The arguments are difficult to follow without going into the details of Albert's original paper.\}
Reviewed by Holger P. Petersson
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The author determines the number $J\sb {l\sp n}(D)$ of elementary abelian $l$-extensions of degree $l\sp n$ over ${Q}$ with discriminant $D$, where $l$ is an odd prime. Furthermore he proves that the number of such fields with conductor at most $x$ and $l$ unramified (or ramified, respectively) equals $c·x·(\log x)\sp {\tau\sb n-1}+O(x·(\log x)\sp {\tau\sb n-2})$ with $\tau\sb n=(l\sp n-1)(l-1)\sp {-1}$ and the explicitly given constant $c$ depending on whether $l$ is unramified or not. Analogous results for $l=2$ are quoted from another paper of the author [same journal A 27 (1984), no. 4, 345--351; MR 86d:11096]. The proof uses Kummer theory, which makes it possible to obtain $J\sb {l\sp n}(D)$ by counting subspaces of appropriate vector spaces over ${F}\sb l$. The second result is proved with Dirichlet series.
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For an abelian extension of the rational number field, the genus field is constructed by means of Hilbert ramification theory without using class field theory. The result contains a correction of some mistakes in the result of M. Bhaskaran [J. Number Theory 11 (1979), no. 4, 488--497; MR 80j:12002].
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Let $J\sb n(d)$ denote the number of fields of degree $2\sp n$ over ${Q}$ of discriminant $d$ which are composites of quadratic fields. Let $N\sb n(X)$ denote the number of such fields of discriminant at most $X$. The problem is to determine $J\sb n(d)$ and $N\sb n(X)$ or at least bounds for them. For $n=2$ and 3 this was accomplished by A. M. Baily [J. Reine Angew. Math. 315 (1980), 190--210; MR 81c:12006; ibid. 328 (1981), 33--38; MR 83d:12002]. In this paper the author, for general $n$, gives $J\sb n(d)$ explicitly and shows $$N\sb n(X)=a\sb nX\sp {2\sp {1-n}}(\log X)\sp {2\sp n-2}+O(X\sp {2\sp {1-n}}(\log X)\sp {2\sp n-3}),$$ where $a\sb n$ is given explicitly. He also corrects some computations in Baily's articles. The proofs depend on the structure of such fields presented by the author in a previous paper [J. China Univ. Sci. Tech. 12 (1982), no. 4, 29--41; MR 84j:12007] which is not available in English.
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In the 1930 paper cited in the title, Albert classified cyclic quartic extensions of the rationals and gave explicit normal bases for them. The author of the paper under review claims that of the 16 fields for which normal bases were given, nine are incorrect. In particular, he shows that two of the cases in Albert's paper (Theorem 12) are impossible, and he obtains a different classification than that given in Albert's Section 8.
As with the original paper, these results are obtained by strictly elementary---but involved---arithmetic arguments.
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Let $k$ be a real quadratic field. By using the ideal class group of $k$ and its fundamental unit, the author constructs the cyclic quartic extensions $K/{Q}$ containing $k$. He then deduces a criterion for a field $K$ to have a relative integral basis over $k$, which improves a result of H. Edgar and B. Peterson [same journal 12 (1980), no. 1, 77--83; MR 891m:12007]. Such a basis is provided whenever it exists. In an appendix the author explicitly computes the discriminant of $K$. As he himself notes, he thus gets a classification of the fields $K$ which is different from that of A. A. Albert [Ann. of Math. (2) 31 (1930), 381--418; Jbuch 56, 870].
Finally, for the paper under review as well as for that of Edgar and Peterson [op. cit.], we have to recall the work of E. Witt [J. Reine Angew. Math. 174 (1936), 237--245; Jbuch 62, 110] in which he constructed in a purely algebraic manner, among other extensions, the cyclic quartic fields over any base field $R$ of characteristic $\neq2$ (see p. 243); moreover the reviewer has also generalized this construction of Witt.
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Let $k$ be a real quadratic number field. The authors exhibit all quadratic extensions $K$ of $k$ such that (1) $K$ is cyclic over ${Q}$ and (2) the ring of integers $O\sb K$ is free over $O\sb k$. Such extensions $K$ exist if and only if the fundamental unit of $k$ has norm $-1$. The authors' theorem explains an observation that has been made previously in the literature. Namely, if $k={Q}(\sqrt{d})$, $d$ a prime, then each $K$ with property (1) also has property (2). This is no longer true if $d$ is composite.
The proof is given in two separate papers in Chinese.
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For the number field of type $(2,2,\cdots,2)$ ($n$ times), which is composed of $n$ quadratic fields over the field of rational numbers, the author gives (a) the decompositions of all the rational primes, (b) all the prime ideals explicitly, (c) the discriminant, different and conductor, (d) an integral basis and (e) a formula for the class number. The same type of formula for the class number of the number field of type $(l,l,\cdots,l)$ ($n$ times; $l$ is a rational prime) was previously given by S. Kuroda [Nagoya Math. J. 1 (1950), 1--10; MR 12, 593]. Another integral basis was given by H. Iwata [Sugaku 24 (1972), 312--314; MR 58 #27896]. The results of the present paper are too long to quote here.
Reviewed by Hong Wen Lu
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This paper presents analysis of the performance of various types of error detecting codes on feedback and retransmission error control systems. The codes under study are some cyclic codes, convolutional codes, and matrix codes. The transmission channel is binary symmetric and is assumed to be time-discrete and memoryless. No error is assumed in the feedback channel. A table is compiled which compares the transmission rates and the output error probabilities of those codes.
Reviewed by Tai Yang Hwang
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CMP 734 820
(16:09) 12A30{There will be no review of this item.}
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