SectionGroups
... Example 9: Determine if Z m is an abelian group under the addition operator. Solution: For a, b, c Z m , we show it satisfies the 4 properties for a group. To do this, let the binary operation m denote the addition modulo m (for example, a m b can be thought of computationally as finding the i ...
... Example 9: Determine if Z m is an abelian group under the addition operator. Solution: For a, b, c Z m , we show it satisfies the 4 properties for a group. To do this, let the binary operation m denote the addition modulo m (for example, a m b can be thought of computationally as finding the i ...
Solutions to Exercises for Section 6
... The last part is asking whether there is a root of Y 2 + 1 = 0 in K; that is, is there an element of K whose square is −1(= 2)? One possibility is just to start checking, squaring the elements of K in turn, to see if 2 is a square in K. Alternatively, take a typical element aα + b, square it and rea ...
... The last part is asking whether there is a root of Y 2 + 1 = 0 in K; that is, is there an element of K whose square is −1(= 2)? One possibility is just to start checking, squaring the elements of K in turn, to see if 2 is a square in K. Alternatively, take a typical element aα + b, square it and rea ...
Modules Over Principal Ideal Domains
... Corollary 12. Each finitely generated abelian group is a direct sum of cyclic groups , each of prime power order or infinite. Proof. This quickly follows from the fact that Z is a PID. So in the particular case where R in Theorem 11 is Z, our Z-module is the same from Example 1, where scaler multipl ...
... Corollary 12. Each finitely generated abelian group is a direct sum of cyclic groups , each of prime power order or infinite. Proof. This quickly follows from the fact that Z is a PID. So in the particular case where R in Theorem 11 is Z, our Z-module is the same from Example 1, where scaler multipl ...
