reciprocal in ordered field

**particlejohn** · July 3rd 2008, 06:20 PM

Let $\mathbb{F}$ be an ordered field and $a \in \mathbb{F}$ . If $a > 0$ , show that $a^{-1} > 0$ . If $a < 0$ , show that $a^{-1} < 0$ .

So let $a,b,c \in \mathbb{F}$ . If $ab = c$ and any two of $a,b, \ \text{or} \ c$ is positive, then so is the third.

Take $b = a^{-1}$ and the Corollary is shown?

Is this correct? Generally, with corollaries, you invoke the theorem?

**Isomorphism** · July 3rd 2008, 11:02 PM

Originally Posted by particlejohn

Let $\mathbb{F}$ be an ordered field and $a \in \mathbb{F}$ . If $a > 0$ , show that $a^{-1} > 0$ . If $a < 0$ , show that $a^{-1} < 0$ .

So let $a,b,c \in \mathbb{F}$ . If $ab = c$ and any two of $a,b, \ \text{or} \ c$ is positive, then so is the third.

Take $b = a^{-1}$ and the Corollary is shown?

Is this correct? Generally, with corollaries, you invoke the theorem?

Well thats correct (since it should be easy to establish 1 > 0 by your axioms)

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