- #1

ehrenfest

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**[SOLVED] easy algebra problem**

## Homework Statement

Why is the following equation impossible:

[tex]\sqrt{3}=a+b\sqrt{2}[/tex]

where a and b are rational numbers and b is not 0. It seems so obvious... Feel free to use group, ring, or field theory in your answer.

## Homework Equations

## The Attempt at a Solution

EDIT: I can prove that [tex]\sqrt{3}[/tex] is irrational. Square both sides and rearrange to get

[tex]\frac{3-a^2-2b^2}{2ab}=\sqrt{2} [/tex]

which is impossible because the rationals form a field (which is closed under addition, multiplication, and division).

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