College of the Holy Cross, Fall 2009 Math 351 Homework 11 selected solutions Chapter 21, #14. By calculating ( 3 + 2)4 and ( 3 + 2)2 , one sees that ( 3 + 2) satises the polynomial 4 + 2x2 + 25. There are two ways one can show this is the minimal x polyno
College of the Holy Cross, Fall 2009 Math 351 Homework 4 selected solutions
Chapter 16, #10. To show that R[x] and S [x] are isomorphic, we need to dene a mapping from R[x] to S [x] and show that it is a ring homomorphism, 1-1, and onto. What we know is t
College of the Holy Cross, Fall 2009 Math 351 Homework 3 selected solutions
Chapter 14, #28 First one must verify that A = cfw_(3x, y ) | x, y Z, the set of all elements whose rst entry is a multiple of 3 and whose second entry is allowed to be anything,
College of the Holy Cross, Fall 2009 Math 351 Homework 2 selected solutions
Chapter 13, #28a. Since R is an abelian group under addition, we know from the Fundamental Theorem of Finite Abelian groups that under addition R is isomorphic to Z6 . Note that t
College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 2 solutions Prof. Jones
1. Find the splitting eld of f (x) = x3 + x + 1 Z2 [x] over Z2 . First note that f (x) is irreducible over Z2 by Theorem 17.1. Now consider Z2 (), which by Theorem 20.
College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 2 1 Prof. Jones
1. Find the splitting eld of f (x) = x3 + x + 1 Z2 [x] over Z2 . 2. (a) The extension Q( 6 5) is a vector space over Q. Find a basis for it. (b) Let be a root of the polynomia
College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 1 solutions Prof. Jones
1. Let I be an ideal in a ring R, and set N (I ) = cfw_x R : xi = 0 for all i I . a) Show that N (I ) is an ideal in R. Using the ideal test, we must show rst that N (
College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 1 Prof. Jones
1. Let I be an ideal in a ring R, and set N (I ) = cfw_x R : xi = 0 for all i I . a) Show that N (I ) is an ideal in R. b) Show that if R is an integral domain, then either I =
College of the Holy Cross, Fall 2009 Math 351 Solutions to the practice problems for exam 3
#1. Suppose that a is algebraic over Q and f (x) is a polynomial with coecients in Q. Show that Q(f (a) is an algebraic extension of Q. Since f (x) has coecients i
College of the Holy Cross, Spring 2010 Math 352, Midterm 2 Monday, April 19
You may use any results from class or from the text, except for homework exercises unless explicitly stated otherwise. To get full credit for your answers, you must fully explain
College of the Holy Cross, Spring 2010 Math 352, Midterm 1 Solutions Monday, March 22
All the numbered problems on this exam are weighted equally (some have two parts, but that counts as one problem). Choose 6 of the following 7 problems to do, and clearl
College of the Holy Cross, Fall 2009 Math 351 Homework 12 selected solutions Chapter 32, #4. Let E = Q( 2, 5, 7). The problem tells us to assume that Gal(E/Q) = Z2 Z2 Z2 . By the Fundamental Theorem of Galois Theory, a subgroup H Gal/(E/Q) has xed eld EH
College of the Holy Cross, Fall 2009 Math 351 Homework 9 selected solutions
Chapter 20, #20. We will show that F (c) F (ac + b) and also that F (ac + b) F (c), which will solve the problem. To show F (c) F (ac + b), we use the fact that F (c) is the small