Math 145, Midterm Exam, Due May 7 in class.
You may assume that the ground eld is k = C.
Problem 1. (10 points.) Let n 2, and let S = cfw_a1 , . . . , an be a nite set with n elements in A1 .
(i) Show that the quasi-afne set A1 \ S is isomorphic to an af
Math 145, Final Exam.
You may assume that the ground eld is k = C.
1. (Rational points on cubic curves.) Consider the point P (0, 0) on cubic curve C with equation
y 2 + y = x3 x.
(i) Let Q(a, b) be a point on the cubic. Show that the inverse Q has coordi
Math 145, Problem Set 7. Due Friday, May 30.
You may assume that the ground eld is k = C.
1. (Etymology.) Part A - Arclength of the ellipse and elliptic integrals. Let 0 < a < b. Consider the
ellipse
x2
y2
+ 2 = 1.
2
a
b
(i) Show that the arclength of the
Math 145 - Midterm Solutions
Problem 1. (10 points.) Let n 2, and let S = cfw_a1 , . . . , an be a nite set with n elements in A1 .
(i) Show that the quasi-afne set A1 \ S is isomorphic to an afne set. For instance, you may take X to
be the afne algebrai
IRREDUCIBILITY AND DIMENSION
DRAGOS OPREA
1. IRREDUCIBILITY Let us assume that k is an algebraically closed eld. We will study afne algebraic sets in a bit more detail. To begin with, we will break them into smaller pieces which cannot be broken further,
HILBERTS NULLSTELLENSATZ
DRAGOS OPREA
1. INTRODUCTION Let k be an algebraically closed eld. We will employ the following notation. If I k[X1 , . . . , Xn ] is an ideal, we let Z(I) denote the afne algebraic set in An dened by the vanishing of the polynomi
Math 145, Problem Set 1.
1. A topological space X is said to be Noetherian if it satises the ascending chain
condition on open sets (i.e. any ascending chain of open sets eventually stabilizes).
Check that any subset Y X is also Noetherian in the subspace
Math 140B - Homework 6. Due Wednesday, May 29.
1. Rudin, Chapter 7, solve problems 20, 21, 23.
2. Show that power series can be integrated term by term within the radius of convergence.
That is, assume that
cn xn
f (x) =
n=0
has radius of convergence R >