Question

19. The quotient field of the integral domain Z of integers is the field of
(A) complex numbers
(B) rational numbers
(C) real numbers
(D) None of these​

Answer 1

Answer:

a. complex number and none of these

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

The quotient field of the integral domain Z of integers is the field of

(A) Complex numbers

(B) Rational numbers

(C) Real numbers

(D) None of these

EVALUATION

Let F be the field of quotients.

$\displaystyle \sf{An \: element \: of \: F \: is \: of \: the \: form \: \: \frac{m}{n} }$

$\sf{Where \: m \in \: \mathbb{ Z} \: , n \in \mathbb{ Z} \: \: and \: \: n \ne \: 0}$

i.e every element of F is a rational number

$\sf{Therefore \: \: \: F \subset \mathbb{Q}}$

$\sf{Where \: \mathbb{Q} \: \: is \: field \: of \: rational \: numbers }$

$\sf{Let \: \: x \in \mathbb{Q}}$

$\displaystyle \sf{Then \: \: x = \frac{p}{q} }$

$\sf{Where \: p \in \: \mathbb{ Z} \: , q \in \mathbb{ Z} \: \: and \: \: q \ne \: 0}$

$\sf{This \: shows \: that \: \: x \in \: F \: }$

$\sf{Therefore \: \: \: \mathbb{Q} \subset F}$

$\sf{Consequently \: \: \: F = \mathbb{Q}}$

FINAL ANSWER

The quotient field of the integral domain Z of integers is the field of

(B) Rational numbers

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