Question

can someone please help me in solving this theorem:
if an integral domain D is of finite characteristic then its characteristic is a prime no. prove it

Answer 1

*

Answer 2

what is this???

Answer 3

bookmark

Answer 4

Relishing the abstract algebra :D

Answer 5

those sora question came in my exam ... i couldn't do it.

Answer 6

I'm thinking...

Answer 7

can u help me please in solving it???

Answer 8

I will, as soon as I can figure it out :D

Answer 9

this is from Ring theory....

Answer 10

currently I am eating ... looks like there's something in proofwiki
http://www.proofwiki.org/wiki/Characteristic_of_Integral_Domain

Answer 11

Aww... a spoiler alert? :P

Answer 12

don't worry ... i won't understand it, i had trouble dealing with simple group theory.

Answer 13

no in ur link finite characteristic condition is not given

Answer 14

Ok... I got it :)

Answer 15

By your will @rosy
At your signal, I proceed :)

Answer 16

looks like she went offline ... post if you like ... even if i don't understand.

Answer 17

Okay, let the characteristic be k, and let the unity = 1.
The unity exists, as this is an Integral Domain.

Then...
k 1 = 0, by definition.
Suppose k = mn, where m and n are less than k (in other words, k is not prime)
then
mn1 = 0
m1n1=0
(1+1+1...)(1+1+1....)=0
m times^ n times^
This is a contradiction, as we have an integral domain, which is not supposed to have zero divisors.

Hence, it can't have been the case that k = mn, where m and n are less than k
In other words, k must have been prime
QED

Answer 18

are u sure in an integral domain unity always exists.......bcz in the def of integrl dmin ,which i study,existence of unity is not mentioned..

Answer 19

Yes, the existence of unity is assumed in an Integral Domain.

Answer 20

ok thanks a lot.....u have solved a major problem of mine....:)