- #1

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I'm just looking for a simple counterexample. I'm working through Herstein's (Topics..) Ring chapter and this is the problem set after chapter 3.2.

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- Thread starter SiddharthM
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- #1

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I'm just looking for a simple counterexample. I'm working through Herstein's (Topics..) Ring chapter and this is the problem set after chapter 3.2.

- #2

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Think about, for example, polynomial rings.

- #3

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The idea of a polynomial ring was not introduced before this problem set. What does the variable X traditionally vary over? Does it matter?

- #4

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How about an algebraic closure of F_p? By the way, why are you looking for a *counter*-example?

- #5

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An indeterminate, ok.

I presume F_p is the field of 0,1,...,p-1 where addition is modp and similarly for multiplication. If you could elaborate as to 1. what the algebraic closure of F_p is, 2. how it is infinite and 3. how it has finite characteristic.

I haven't been formally introduced to algebraic closures or polynomial rings as yet in Herstein yet I'm expected to come up with this example. There must be something simpler...or maybe not.

Thanks for the help.

- #6

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It is easy to show that no field F_q where q=p^r is algebraically closed (you should try this - hint the non zero elements a a group of order p^r - 1).

Thus if such a thing as an algebraic closure exists, it is infinite.

Anyway, that is probably beyond what you were expected to know.

Since we don't necessarily know what is in Herstein, we cannot say what you're supposed to know or not, but the polynomial ring idea should have been something you can come up with. Just because it is not formally introduced yet doesn't mean you weren't supposed to come up with something like that. After all, you knew what polynomials are before starting book, right?

- #7

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word

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