104_Fa08_hw_1_sol

# 104_Fa08_hw_1_sol - Homework 1 for MATH 104 Brief Solutions...

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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 1 for MATH 104 Brief Solutions Problem 1 (a) Let F be an ordered field. Let x F . Show that if x &gt; 0 then- x &lt; , if x , 0 then x 2 &gt; . Solution. If 0 &lt; x , then by axiom (OF1), 0 + (- x ) &lt; x + (- x ), hence- x &lt; 0. For the second assertion, we first show that (- a ) 2 = a 2 . Note that = a = a ((- a ) + a ) = a (- a ) + aa , hence a (- a ) =- aa . We use this to infer (- a )(- a ) + (- aa ) = (- a )(- a ) + (- a ) a = (- a )((- a ) + a ) = (- a )0 = , and therefore (- a )(- a ) = aa . Now, if a &gt; 0, then aa &gt; a = 0 by (OF2). If a &lt; 0, then an argument analogues to the first part show that- a &gt; 0. Then 0 &lt; (- a )(- a ) = a 2 . (b) On the set R 2 define the following operations of addition and multiplication: ( x 1 , x 2 ) + ( y 1 , y 2 ) = ( x 1 + y 1 , x 2 + y 2 ) ( x 1 , x 2 )( y 1 , y 2 ) = ( x 1 y 1- x 2 y 2 , x 1 y 2 + x 2 y 1 ) This turns R 2 into a field. (Note: this is just the field of complex numbers, if one identifies a complex number with an element of R 2 , its real and imaginary part.) Show that no order can be defined on this field that turns it into an ordered field....
View Full Document

## 104_Fa08_hw_1_sol - Homework 1 for MATH 104 Brief Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online