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AAP Interview: Wah-Ming Chang
Get PDFWorra, Bryan Thao. (2004). AAP Interview: Wah-Ming Chang. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/166378
Retract rational fields
No full textAbstractLet k be an infinite field. The notion of retract k-rationality was introduced by Saltman in the study of Noetherʼs problem and other rationality problems. We will investigate the retract rationality of a field in this paper. Theorem 1: Let k⊂K⊂L be fields. If K is retract k-rational and L is retract K-rational, then L is retract k-rational. Theorem 2: For any finite group G containing an abelian normal subgroup H such that G/H is a cyclic group, for any complex representation G→GL(V), the fixed field C(V)G is retract C-rational. Theorem 3: If G is a finite group, then all the Sylow subgroups of G are cyclic if and only if Cα(M)G is retract C-rational for all G-lattices M, for all short exact sequences α:0→C×→Mα→M→0. Because the unramified Brauer group of a retract C-rational field is trivial, Theorems 2 and 3 generalize previous results of Bogomolov and Barge respectively (see Theorems 5.9 and 6.1)
Noether's problem for p-groups with a cyclic subgroup of index p2
No full textAbstractLet K be any field and G be a finite group. Let G act on the rational function field K(xg:g∈G) by K-automorphisms defined by g⋅xh=xgh for any g,h∈G. Noether's problem asks whether the fixed field K(G)=K(xg:g∈G)G is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order pn (n⩾3) containing a cyclic subgroup of index p2 and K is any field containing a primitive pn−2-th root of unity, then K(G) is rational over K. As a corollary, if G is a non-abelian p-group of order p3 and K is a field containing a primitive p-th root of unity, then K(G) is rational
Bezout's Theorem and ideals of terminal forms
No full textAbstractLet k be any field, k[X1,…,Xn] be the polynomial ring of n variables over k. For any f=f0+f1+⋯+fr∈k[X1,…,Xn] where each fi is a homogeneous polynomial of degree i and fr≠0, define tm(f)=fr. If I is an ideal in k[X1,…,Xn], define tm(I) to be 〈tm(f):f∈I⧹{0}〉, the ideal generated by the terminal forms tm(f). Using Bezout's Theorem and Macaulay's Theorem, we will establish the following. If f,g∈k[X1,X2] satisfying that gcd{f,g}=gcd{tm(f),tm(g)}=1 and I=〈f,g〉, then tm(I)=〈tm(f),tm(g)〉. Actually the above result is equivalent to Bezout's Theorem, which sheds another perspective of Bezout's Theorem. These results are valid in k[X1,…,Xn] also
Noether's problem for metacyclic p-groups
No full textAbstractLet K be any field and G be a finite group. Let G act on the rational function field K(xg:g∈G) by K-automorphisms defined by g·xh=xgh for any g,h∈G. Denote by K(G) the fixed field K(xg:g∈G)G. Noether's problem asks whether K(G) is rational (=purely transcendental) over K. An affirmative answer to Noether's problem for metacyclic p-groups will be proved provided that K contains enough roots of unity
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