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Topics In Algebra Elementary Algebraic Geometry

Môn học gồm bốn chương. Chương 0 cung cấp cho người học những hiểu biết sơ lược về nhóm, vành, trường, . đủ để hiểu được các chương tiếp theo. Chương 1 và chương 2 bước đầu tiếp cận ngôn ngữ trừu tượng về không gian vectơ và ánh xạ tuyến tính. Chương 3 giới thiệu những khái niệm quan trọng của Đại số tuyến tính như định thức, hạng của ma trận | 2003 Topics In Algebra Elementary Algebraic Geometry David Marker Topics In Algebra Elementary Algebraic Geometry David Marker Spring 2003 Contents 1 Algebraically Closed Fields 2 2 A ne Lines and Conics 14 3 Projective Space 23 4 Irreducible Components 40 5 Bezout s Theorem 51 1 Let F be a field and suppose f1 . fm 2 F X1 Xn . A central problems of mathematics is to study the solutions to systems of polynomial equations fi X1 Xn 0 f2 X1 Xn 0 . . . fm X1 Xn 0 where f1 fm 2 F X1 Xn . Of particular interest are the cases when F is the field Q of rational numbers R of real numbers C of complex numbers or a finite field like Zp. For example Fermat s Last Theorem is the assertion that if x y z 2 Q n 2 and xn yn zn then at least one of x y z is zero. When we look at the solution to systems of polynomials over R or C we can consider the geometry of the solution set in Rn or Cn . For example the solutions to X2 - Y2 1 is a hyperbola. There are many questions we can ask about the solution space. For example i The circle X2 Y2 1 is smooth while the curve Y2 X3 has a cusp at 0 0 . How can we tell if the solution set is smooth ii If f g 2 C X Y how many solutions are there to the system f X Y 0 g X Y 0 The main theme of the course will be that there are deep connections between the geometry of the solution sets and algebraic properties of the polynomial rings. 1 Algebraically Closed Fields We will primarily be considering solutions to f X Y 0 where f is a polynomial in two variables but we start by looking at equations f X 0 in a single variable. In general if f 2 F X there is no reason to believe that f X 0 has a solution in F. For example X2 2 0 has no solution in Q and X2 1 0 has no solution in R. The fields where every nonconstant polynomial has a solution play an important role. Definition 1.1 We say that a field F is algebraically closed if every nonconstant polynomial has a zero in F.