# Tagged Questions

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### Does $x\cdot 0 = 0$ follow from the field axioms alone?

From the field axioms alone, does it follow that $x \cdot 0 = 0$ for all $x$? All I would like is a statement that it can or cannot be done (hints not necessary). I would like to do it myself; I ...
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### Using field axioms for a simple proof

Question: If $F$ is a field, and $a, b, c \in F$, then prove that if $a+b = a+c$, then $b=c$ by using the axioms for a field. Relevant information: Field Axioms (for $a, b, c \in F$): ...
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### One question about additive identity arising from Apostol's field axiom in the Mathematical Analysis 2nd edition

In the begining of this book, the field axiom 4 talks about "Given any two real numbers x and y, there exists a real number z such that x+z=y and this z is denoted by y-x." Therefore, for each x, we ...
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### Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
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### Real numbers axiomatization without natural numbers

I think to remember that there is way to uniquely characterize the real numbers $\mathbb{R}$ via an axiom set. I wonder if this is possibly without introducing some notion of the natural numbers ...
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### Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
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### Proving $1 > 0$ using only the field axioms and order axioms

How do I prove $1 > 0$ using only field axioms and order axioms? I have tried using the cancellation law, with the identities in a field and I cannot get anywhere. Does anybody have any ...
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### Why would the author ask if I used the Associative Law to prove + is not equiv. to *?

I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows: Problem 1: Show that + and * are necessarily different operations. That is, for any ...
### Show $xy\neq0$ is the same as $x\neq0 \wedge y \neq0$
I have to show: $$xy\neq0 \Leftrightarrow x\neq0 \wedge y \neq0$$ I think I can "simplify" it to this: $$xy=0 \Leftrightarrow x=0 \vee y=0$$ Since $a\cdot0=0$ is an proven theorem, I can show: ...