1
vote
2answers
88 views

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 ...
3
votes
3answers
163 views

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 $a, b, c \in F$): ...
1
vote
3answers
55 views

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 ...
23
votes
6answers
1k views

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. ...
1
vote
1answer
85 views

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 ...
21
votes
6answers
886 views

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). ...
3
votes
1answer
657 views

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 ...
5
votes
4answers
163 views

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 ...
2
votes
3answers
110 views

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: ...
1
vote
6answers
607 views

Is it possible to have a field without an additive identity?

If I drop the axiom that Zero is the identity of an addition what consequences does this entail? What do I need to change to my axiomatization? By definition it is not possible, but are there ...