GEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS JENNYWANG Abstract. In this paper, we study eld extensions obtained by polynomial rings and maximal ideals in order to determine whether solutions exist to three ancient Greek construction problems: squaring the circle, doubling the cube, and trisecting an angle. Contents 1. Introduction 1 2. Principal Ideal Domains and Polynomial Division 2 3. Field Extensions 3 4. Algebraic Extensions 6 5. Impossibility of Geometric Constructions 8 5.1. Squaring the Circle 10 5.2. Doubling the Cube 11 5.3. Trisecting an Angle 11 6. Acknowledgments 12 References 12 1. Introduction Much of Ancient Greek ...

 

Problem of the Month: Cutting a Cube The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: “Make sense of problems and persevere in solving them.” The POM may be used by a teacher to promote problem solving, and to address the differentiated needs of her students. A department or grade level may engage their students in a POM to showcase problem solving as a key aspect of doing mathematics. POMs can also be used schoolwide ...