Explanation Given information: D is a field. Formula used: 1) Definition of ring isomorphism: Let R and R ' denote two rings. A mapping ϕ : R → R ' is a ring isomorphism from R to R ' provided the following conditions hold: 1. ϕ is one-to-one correspondence (a bijection) from R to R ' . 2. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) for all x and y in R . 3. ϕ ( x ⋅ y ) = ϕ ( x ) ⋅ ϕ ( y ) for all x and y in R . 2) The multiplication of ordered pairs is given by ( x , y ) ⋅ ( z , w ) = ( x z − y w , x w + y z ) 3) The addition of ordered pairs is given by ( x , y ) + ( z , w ) = ( x + z , y + w ) Proof: Define a mapping ϕ : Q → D such that ϕ ( [ a , b ] ) = a b − 1 . A mapping ϕ : Q → D is an isomorphism from Q to D if it satisfies the following conditions: Let us first check if ϕ is well defined: Suppose [ a , b ] = [ c , d ] ⇒ a d = b c . Pre multiply by b − 1 and post multiply by d − 1 on both sides. b − 1 a d d − 1 = b − 1 b c d − 1 . This implies b − 1 a = c d − 1 . By commutativity, a b − 1 = c d − 1 . ϕ [ a , b ] = ϕ [ c , d ] . Therefore, ϕ is well defined. 1. ϕ is one-to-one correspondence (a bijection) from Q to D : Let ϕ [ a , b ] = ϕ [ c , d ] So, a b − 1 = c d − 1 By using commutativity, b − 1 a = c d − 1 Pre multiply by b and post multiply by d on both sides. b b − 1 a d = b c d − 1 d a d = b c ⇒ [ a , b ] = [ c , d ] Therefore, ϕ is one-to-one. Let d ∈ D ; then there exists [ d , e ] ∈ Q such that ϕ ( [ d , e ] ) = d . Therefore, ϕ is onto. 2. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) for all x and y in Q : Let [ a , b ] , [ c , d ] ∈ Q Let ϕ ( [ a , b ] + [ c , d ] ) By using addition in Q , = ϕ ( [ a d + b c , b d ] )

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Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 5.3, Problem 12E

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To prove: If D is a field, then the field of quotient Q is isomorphic to D.

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To prove: If D is a field, then the field of quotient Q is isomorphic to D .

Solution Summary: The author explains that if D is a field, then the field of quotient Q would be isomorphic to it. A mapping varphi :Rto

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To prove: If D is a field, then the field of quotient Q is isomorphic to D.

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