Math Is Fun Forum

Zhivago

Member

Registered: 2008-11-21

Posts: 5

Field Axioms

By definition, do the Field Axioms of the real number system apply to the zero (0) and the unit/identity (1) elements too?  I'm asking this because there is no element for the zero element with respect to the operation of multiplication that will yield the identity element (1).  I know I'm missing something somewhere.  I'll really appreciate if someone can take out the time to provide some clarification.

Thanks,
Zhivago...

Zhivago

Member

Registered: 2008-11-21

Posts: 5

Re: Field Axioms

Such an idiot I'm!  Got what I was missing   Sorry for the trouble guys...

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Field Axioms

I know you got it, but just to be certain, for any field F, F* = F - {0} is a multiplicative group.

There is a small implication that comes out of this.  {0} is not a field, since {0} - {0} is the empty set, which is not a group.  That means there must be at least two elements in a field F, in particular, 0 and 1.  It also turns out that the characteristic of a field (the order of the additive group generated by 1) must be a prime number.  Many people take this as saying that 1 is not prime.

Of course, there are much better reasons for 1 not being prime.  This is just a little bit of icing.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Zhivago

Member

Registered: 2008-11-21

Posts: 5

Re: Field Axioms

What I got was that - the field axiom regarding the existence of multiplicative inverse doesn't apply to the zero element.

I didn't know anything about multiplicative and additive groups, but after reading about them I can make sense of whatever you said.  But what I didn't get is the link between multiplicative group and my question.

I was assuming that a field needs to have at least two elements because of the presence of two binary operations.  I guess that was a totally wrong one.  What you said makes the perfect sense now, because one needs to consider the existence of the zero and identity elements too.  Right?!

I'm trying to lay a very strong foundation before I jump into calculus and learn its concepts.  It seems like I will need a firm grounding in set theory, real number system, and proof by mathematical induction.  Right now, I'm looking at the real  number system using the books by T.M. Apostol (Calculus I) & R.G. Bartle (Elements of Real Analysis).  Can you suggest anything else that would be good?

Thanks Ricky!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Field Axioms

I'm trying to lay a very strong foundation before I jump into calculus and learn its concepts.

What country are you in?  In America, students learn calculus and then many years down the road learn real analysis.  From what I've heard from several foreign friends is that the rest of the world learns both at the same time.

Also, I thought you'd be familiar with groups and rings.  If you aren't, then ignore what I said before.  In a field, 0 can not have a multiplicative inverse.  Here is a proof of that:

Let F be a field.  First, we establish that a*0 = 0 for all a in F.  We have that 0 + 0 = 0, and left multiplying by a gives a(0 + 0) = a*0.  Using the distributive property, a0 + a0 = a0.  Now we subtract a0 from both sides, so a0 + a0 - a0 = a0 - a0.  a0 - a0 = 0 (additive inverse), and so a0 = 0.

Now let's go find a multiplicative inverse of 0.  We want to find some x such that x0 = 1.  But we've already shown that for all x, x0 = 0, and it must be that 0 is not equal to 1.  Therefore, no such x exists, and we conclude that 0 can not have a multiplicative inverse.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Zhivago

Member

Registered: 2008-11-21

Posts: 5

Re: Field Axioms

I'm in India; have a bachelor's degree in engineering, and it's been a long time since I graduated (1997).  I have gone through the calculus courses in college, but not real analysis.  And to be frank, I never concentrated on mathematics.  Now I'm thinking of pursuing higher studies, and I want to do some self-study and improve my mathematical skills.  So, I want to get really good at calculus, linear algebra, and probability theory so that I can apply them with ease in my higher studies. I just started brushing up the basics.

I'm familiar with the concepts of algebraic structures, but never delved into it.  What you said makes sense to me now.

Thanks a bunch Ricky!