Explanation Given information: F denotes the field of quotients of the ring of even integers E . 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: Given that F denotes the field of quotients of the ring of even integers E . Define a mapping ϕ : Q → F such that ϕ ( [ m , n ] ) = [ 2 m , 2 n ] . A mapping ϕ : Q → F is an isomorphism from Q to F if it satisfies the following conditions: Let us first check if ϕ is well defined: Suppose [ m 1 , n 1 ] = [ m 2 , n 2 ] ⇒ m 1 n 2 = n 1 m 2 . Multiply both sides by 4 . ⇒ 4 m 1 n 2 = 4 n 1 m 2 ⇒ [ 2 m 1 , 2 n 1 ] = [ 2 m 2 , 2 n 2 ] ⇒ ϕ ( [ m 1 , n 1 ] ) = ϕ ( [ m 2 , n 2 ] ) . Therefore, ϕ is well defined. 1. ϕ is one-to-one correspondence (a bijection) from Q to F : Let ϕ [ m 1 , n 1 ] = ϕ [ m 2 , n 2 ] So, [ 2 m 1 , 2 n 1 ] = [ 2 m 2 , 2 n 2 ] ⇒ 4 m 1 n 2 = 4 m 2 n 1 ⇒ m 1 n 2 = n 1 m 2 ⇒ [ m 1 , n 1 ] = [ m 2 , n 2 ] Therefore, ϕ is one-to-one. Let [ 2 m , 2 n ] ∈ F ; then there exists [ m , n ] ∈ Q such that ϕ ( [ m , n ] ) = [ 2 m , 2 n ] . Therefore, ϕ is onto. 2. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) for all x and y in Q : Let [ m 1 , n 1 ] , [ m 2 , n 2 ] ∈ Q . Let ϕ ( [ m 1 , n 1 ] + [ m 2 , n 2 ] )

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Chapter 5.3, Problem 13E

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To prove: F is isomorphic to the field of rational numbers where F denotes the field of quotients of the ring of even integers E.

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Chapter 5.3, Problem 12E

Chapter 5.3, Problem 14E

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To prove: F is isomorphic to the field of rational numbers where F denotes the field of quotients of the ring of even integers E .

Solution Summary: The author explains that F denotes the field of quotients of the ring of even integers E.

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To prove: F is isomorphic to the field of rational numbers where F denotes the field of quotients of the ring of even integers E.

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