- #1

VrhoZna

- 14

- 3

## Homework Statement

Prove that each subfield of the field of complex numbers contains every rational number. '

From Hoffman and Kunze's Linear Algebra Chapter 1 Section 2

## Homework Equations

## The Attempt at a Solution

Suppose there was a subfield of the complex numbers that did not contain every rational number (from now on referred to as F), that is there is a rational number p/q, where p and q denote integers, that is not an element of F. Then it follows that either p ∉ F or 1/q ∉ F (as their product is not an element.) We consider each case separately.

Suppose p ∉ F, then (p - 1) ∉ F as (p - 1 + 1 = p) and similarly (p - 2) ∉ F, we proceed stepwise and find that p - (p - 1) ∉ F but of course (p - (p - 1)) = 1 contradicting our assumption that F is a subfield of the complex numbers.

Now suppose 1/q ∉ F, then q ∉ F (as there would be no element x such that x*q = 1) and a similar argument as above finds 1 ∉ F contradicting our assumption that F is a subfield of the complex numbers. Thus every subfield of the complex numbers contains as elements every rational number.

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I feel my reasoning is correct but given that my knowledge of fields is limited to that narrow introduction in the section I'm not sure if any misunderstandings on my part have cropped up.