why isnt Z mod 6 a field?

this is an extra credit problem in my pre-calc class...

I know that it must abide to the field axioms(am i right?)

but mostly i dont understand the form Z mod 6

Printable View

- December 20th 2006, 07:17 PMtummblerz mod 6
why isnt Z mod 6 a field?

this is an extra credit problem in my pre-calc class...

I know that it must abide to the field axioms(am i right?)

but mostly i dont understand the form Z mod 6 - December 20th 2006, 07:30 PMThePerfectHacker
Okay, the basic answer is that it is not a power of a prime. But that is a more advanced answer.

By you can do it by checking whether it is a field or not.

The set is,

It froms a group under addition.

Furthermore, it is a ring (just check the definitions for a ring).

Furthermore, it is a commutative ring.

Furthermore, it has unity (1).

Thus, it is a commutative ring with unity.

Now, if every non-zero element has an inverse the proof is complete.

Thus, we need to show every element has a multiplicative inverse.

The element 2, has no inverse!

Just check each one and convince thyself. - December 20th 2006, 07:42 PMtummbler
what parts of the set are a and b..

im sorry if i sound stupid. ive just neever seen this type of problem before - December 20th 2006, 07:45 PMThePerfectHacker
- December 20th 2006, 07:49 PMtummbler
this is a extra credit prob for precalc.

in the definition of the addition field axiom is has the variables a and b. same in alot of the definitions of the terms u used in your original explanation. I used mathworld.com btw.. - December 20th 2006, 07:53 PMThePerfectHacker
It is not a fair question for a pre-calculus class.

Just write on your paper, that not all elements (numbers in Z6) has an inverse, for example 2 is in Z6 and it got no inverse.

Inverse means we need to find a number 'x' (in Z6) such that,

2x=1

Now, just check all the possibilities,

2(0)=0

2(1)=2

2(2)=4

2(3)=6=0 (because we take the mod of 6)

2(4)=8=2 (same reason)

2(5)=10=4

Note, we went through all possibilities because,

Z6= { 0,1,2,3,4,5 }

And none of them produce 1.