How to prove that $x^{-1}+y^{-1}+z^{-1} \ne 0$?

Here's what's up with $w = xy^{-1}$.

Suppose for the sake of contradiction that $x+y+z=0$ and $x^{-1} + y^{-1} + z^{-1} =0$, or $z = x+y$ and $z^{-1} = x^{-1}+y^{-1}$. Multiplying these together, we get $$1 = zz^{-1} = (x + y)(x^{-1} + y^{-1}) = 1 + x^{-1}y + xy^{-1} + 1 = x^{-1}y + xy^{-1}.$$ If we let $w = xy^{-1}$, this means $w + w^{-1} = 1$, which we can rearrange to get $w^2 + w + 1 = 0$. Multiplying by $w+1$, we have $w^3 + 1 = 0$, or $w^3 = 1$.

But we're given that there are no elements of order $3$, contradiction.

Tags:

Finite Fields

Coding Theory

Elementary Number Theory

Related

Rounding Percents: how far from 100? Generalization of Mathematical Induction? Why does $\epsilon$ come first in the $\epsilon-\delta$ definition of limit? Counter example for limit Why is this seemingly valid argument using modus tollens not valid? How is this an example of "begging the question"? Hermitian Operators and the Spectral Theorem $f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$ Frequently in Nets Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement: What makes a function not defined? What is the expected number of flips of an unfair coin until you have 2 more heads than tails? What is the definition of a first order method?

Recent Posts