Recent content by LorenzoMath

cool ,man! you know numbers, especially prime numbers, do sing! at least, according to a certain "celebrated" mathematician I know. To see the general pattern, write, for example, 9999 as 10000-1 and use distribution law and stuff and calculate. you can see how the first four digits and the...

(\mathbf{Z}/mn\mathbf{Z})^{\times}=(\mathbf{Z}/m\mathbf{Z})^{\times}\times{(\mathbf{Z}/n\mathbf{Z})^{\times}}. Take the orders of both sides. ////

Yes, k[X,Y]=k[X][Y]=k[Y][X]. k[X,Y] is NOT equal to k(Y)[X].

I denoted a ring generated by "cuberoot of 3" over Z by Z[cuberoot of 3]. Let us denote the cube root of 3 by r for the sake of notation. Then the set in question is the same as Z[a] because a^3=3. By the way, in general, if K is a subfield of F and a is an element of F algebraic over...

Note that for a polynomial is of degree 2 and 3, reducibility is equivalent to the existence of roots. mod 3: X^3-23X^2-97X+291=X^3+X^2+2X=X(X^2+X+2). A calculation shows X^2+X+2 doesn't have a root in Z/3Z. Done. mod 7: X^3+5X^2+2X+4. A calculation shows it has no root in Z/7Z. The...

well, so you need to show Z[cuberoot of 3] is a field. Proof 1. Think of the map Q[X] to Z[cuberoot of 3] that maps X to "cuberoot of 3". This is obviously a surjective ring-homomorphism. The kernel is the ideal generated by an irreducible polynomial X^3-3. Thus we have an isomorphism...

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i once heard a guy talking about galois fields. I asked him what the heck it was. he said it is a finite field. So, if a finite field has q elements, then q is a power of some prime p. there is a subfield F_p in it. Z_p=inv.lim. F_p^n.

In arithmetic geometry, one usually uses the letter, p, to denote the characteristic of a base field and "l" for a prime number different from the char. For example, l-adic etale cohomology. p-adic crystalline cohomology.

books if you are studying riemannian geometry, i recommend Manfredo P. do Carmo's Riemannian Geometry. It's a thin book. His definition of connection is axiomatic, but reducing general definitions to down-to-earth definitions or ideas is usually not that hard. It's a nice book.

1234567891011... is not a conventional way of representing real numbers, so unless you introduce your own convention, it doesn't mean anything. Whereas if you put a disimal point somewhere, it represents a real number in a conventional sense. Because, by convention, 1.234567... represents some...

yes, i wanted x to be the generic points of irreducible components of f^(-1)(Z).

I really appreciate your detailed suggestions and comments. What I originally wanted to do was to prove the projection formula of intersection theory, namely f_*(a.f^*b)=f_*a.b. for a flat f. I explicitly wrote down the both sides using Serre's Tor formula. After a little formal manipulation...

I guess I wasn't clear about my question. What I wanted to ask was if we can express \mathcal{O}_X,_{f^{-1}(y)} in terms of \mathcal{O}_X,_x. Here x is a scheme-theoretic point(s). This shouldn't be possible in general, but when X and Y are reduced schemes of finite type, the morphism f is flat...

CORRECTION All the superscripts should be subscripts.