Re: Primitive Elements

On 16 Oct 2005 13:18:17 -0700, jeffconner50@xxxxxxxxx wrote:

>
>Timothy Murphy wrote:
>> jeffconner50@xxxxxxxxx wrote:
>>
>> >> Are you talking about primitive elements in a finite field F_q ?
>> >> If so, an element is primitive iff it has multiplicative order q-1,
>> >> and conjugate elements have the same order (in any group).
>>
>>
>> > Yes, I'm sorry. I failed to mention that it was over F_q. However, I
>> > don't see how the multiplicative order being q-1 for both demonstrates
>> > that the conjugate elements are primitive.
>>
>> Sorry, I was talking nonsense.
>> Or at least, what I said was true but irrelevant in this case.
>> You are not using the word "conjugate" in the group-theoretic sense.
>>
>> There are 2 senses in which you could mean conjugate:
>> 1. Two elements are conjugate if they have the same minimal polynomial;
>> 2. Two elements are conjugate if there is a field automorphism
>> taking one into the other.
>> In this case they are equivalent.
>>
>> With either definition, I would have thought it was fairly easy to see
>> that 2 conjugate elements satisfy the same equations
>> over the prime subfield; so if one had order r (and satisfied x^r = 1)
>> then so would the other.
>>
>> --
>> Timothy Murphy
>> e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
>> tel: +353-86-2336090, +353-1-2842366
>> s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
>
>Sorry, I haven't taken any abstract algebra courses. There is just a
>brief blurb about groups, rings and fields in the books. So I'm not
>really strong when it comes to proofs. The definition in the book is
>the first one, two elements are conjugate if they have the same minimal
>polynomial.
>
>I know that one element could have order r for x^r=1 but what I'm
>missing, for whatever reason, is why all others must have the order r
>as well.
>
>
>Jeff

To even understand the concepts which underlie your question, you
would need to learn some abstract algebra -- groups, rings, fields. If
questions like that intrigue you, then use that as a motivation to
study the theory in depth. Why settle for an explanation based on
concepts you don't yet understand?

quasi
.

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