Comments on: Introduction to Algebraic Geometry (2 in a series) http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-2-in-a-series/ Because my LiveJournal is too silly Fri, 26 Feb 2010 01:22:12 +0100 http://wordpress.org/?v=2.8.6 hourly 1 By: lhrrwcc http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-2-in-a-series/comment-page-1/#comment-86586 lhrrwcc Thu, 06 Mar 2008 22:34:05 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-2-in-a-series/#comment-86586 Another common and important theorem that is worth to mention are the Weak Nullstelensatz, this theorem says that: If an ideal I in the polynomial ring k[X_1,..., X_n] over an algebraic closed field k does not contain the idenity (ie I!=(1)) then V(I) != {}. Also, the corollary of this theorem (in some books this is called the Weak Nullstelensatz) says that: A maximal ideal of the polynomial ring k[X_1,..., X_n] (k is algebraic closed field) has the following form: (X_1 -a_1, ..., X_n - a_n) with a_i in k. The beauty of this is that it gaves us a biyective correspondence between A^n (ie points) and maximal ideals of k[X_1,...,X_n] This, will be clear in the next post Another common and important theorem that is worth to mention are the Weak Nullstelensatz, this theorem says that:

If an ideal I in the polynomial ring k[X_1,..., X_n] over an algebraic closed field k does not contain the idenity (ie I!=(1)) then V(I) != {}.

Also, the corollary of this theorem (in some books this is called the Weak Nullstelensatz) says that:

A maximal ideal of the polynomial ring k[X_1,..., X_n] (k is algebraic closed field) has the following form:
(X_1 -a_1, …, X_n – a_n) with a_i in k.

The beauty of this is that it gaves us a biyective correspondence between
A^n (ie points) and maximal ideals of k[X_1,...,X_n]

This, will be clear in the next post

]]> By: Carnival of Mathematics 1000 « JD2718 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-2-in-a-series/comment-page-1/#comment-82096 Carnival of Mathematics 1000 « JD2718 Fri, 22 Feb 2008 20:27:39 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-2-in-a-series/#comment-82096 [...] 21 - Aaron Roth explores social welfare in Nash equilibrium, but the aspect that catches his attention is the “Price of Malice.” 21 - In Basics of Patch Theory Brent explores the mathematical theory behind version control systems, particularly as it applies to collaborative editing. 21 - Suresh brings a guest blogger in to the Geomblog (Ganesh Gopalakrishnan) to discuss work on model checking by Clarke, Emerson, and Difakis (won the ACM Turing award). 21 - Michi finished his thesis, and found the time to post an introduction to algebraic geometry (in two parts). [...] [...] 21 – Aaron Roth explores social welfare in Nash equilibrium, but the aspect that catches his attention is the “Price of Malice.” 21 – In Basics of Patch Theory Brent explores the mathematical theory behind version control systems, particularly as it applies to collaborative editing. 21 – Suresh brings a guest blogger in to the Geomblog (Ganesh Gopalakrishnan) to discuss work on model checking by Clarke, Emerson, and Difakis (won the ACM Turing award). 21 – Michi finished his thesis, and found the time to post an introduction to algebraic geometry (in two parts). [...]

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