This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1.3 The rational numbers 23 3. x G, ! y G such that x + y = y + x = 0. Write y = x . Written multiplicatively, ( G, , 1) must satisfy 1. x,y G = x y is a welldefined element of G . 2. !1 such that x y = y x = x, x G . 3. x G, ! y G such that x y = y x = 1. Write y = 1 x . Theorem 1.3.2. Z , Q , R , C are groups under addition. N , N are not. Theorem 1.3.3. Let Q = { x Q . . . x 6 = 0 } . Then Q is a group under multiplication. So are R and C . Z is not. Definition 1.3.4. A set which is a group under addition, and whose nonzero elements form a group under multiplication is called a field if the two operations behave nicely together: a ( b + c ) = ( a b ) + ( a c ) (Distributive law) and the operations + , are commutative. Theorem 1.3.5. Q , R , C are fields. GL n = { invertible n n matrices } is not....
View
Full
Document
 Fall '11
 Wong

Click to edit the document details