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''Not sure if this is sound. Comments?'' | ''Not sure if this is sound. Comments?'' | ||
--[[User:Bcaulkin|Bcaulkin]] 21:27, 1 April 2009 (UTC) | --[[User:Bcaulkin|Bcaulkin]] 21:27, 1 April 2009 (UTC) | ||
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| + | So, I'm not entirely sure what angle the question is going at, but I think taking x^3 in Z mod 8Z will work as an example showing that the corollary does not hold. --[[User:Jcromer|Jcromer]] 22:19, 1 April 2009 (UTC) | ||
Revision as of 17:19, 1 April 2009
Corollary 3 of THM 16.2 : "A polynomial of degree n over a field has at most n zeros, counting multiplicity"
Fields and an finite integral domains are one and the same. (THM 13.2)
Finite integral domains are commutative rings with unity and no zero-divisors (Definition of integral domain)
So, if the commutative ring has zero divisors, it cannot be a field, thus no polynomials may over it, thus Corollary 3 is false for any ring with zero-divisors.
Not sure if this is sound. Comments? --Bcaulkin 21:27, 1 April 2009 (UTC)
So, I'm not entirely sure what angle the question is going at, but I think taking x^3 in Z mod 8Z will work as an example showing that the corollary does not hold. --Jcromer 22:19, 1 April 2009 (UTC)
