Primitive elements and order made easy

Solutions to typical test or exam questions.

[@mac2105]

Doesn't Zn usually include zero?

[@RandellHeyman]

Well noticed. I'll pin your comment so others can see it easily.

[@mac2105]

@@RandellHeyman thanks for clarifying

[@LunaOoze]

This was absolutely perfect. After going through quite a few different sources for explainations, this, by far, was the clearest and easiest explaination to help me understand. Thank you for this video.

[@RandellHeyman]

Thanks for taking the time to comment; much appreciated. Other videos at czcams.com/users/randellheyman

[@jb84344]

So glad to have stumbled across this video. I couldn't understand what my books were saying and in less than 8 minutes you've cleared up all the confusion.
Thank you for your very clear explanation. I will be working my way through your other videos!

[@RandellHeyman]

Thanks for the comments. Lots of other videos at www.youtube.com/ randellheyman

[@nathanmartinez2630]

subscribed!

[@eamonnsiocain6454]

Very clear. Thank you.

[@simon_e36]

This was incredibly helpful, truly made easy. Thank you

[@RandellHeyman]

Thanks for such positive comments. Hope some of my other videos are also helpful.

[@werewolf_13]

Brilliant!

[@aibdraco01]

Thank you so much!! I really had hard time understanding the concept! Really helpful!

[@RandellHeyman]

Thanks for letting me know.

[@vidret]

regarding no. 5
if n is anything other than a prime, is there even a primitive element?
Since the totient(n) will be less than n-1 even if an element had this order it wouldn't generate all the elements (since there are still n-1 elements).
Is this correct or am I missing something?
Great videos dude.

[@RandellHeyman]

In some cases where n is a prime power we do have primitive elements. For example, there are primitive elements modulo 4 and 27.
When n is product of more than one prime we also can have a primitive elements. For example, there are primitive elements modulo 6, 10 and 18.
As to your second question/statement phi(10)=4. So the number of elements of Z^*(10) is 4 (i.e. 1,3,7,9). A primitive element will have an order of 4.
Have a look at Primitive root modulo n on Wikipedia for more information.

[@vidret]

@@RandellHeyman dude, yes.

[@waylonjepsen7730]

Great video!

[@RandellHeyman]

Thanks Waylon.

[@PETAJOULE543]

So, primitive element is the element that has to go through all elements before it reaches the identity element that is 1?

[@RandellHeyman]

Yes

[@user-wr5us9ne1n]

i have spent hours unsuccessfully trying to understand this subject and u explained perfect in less than 8 min, great job and ty

[@RandellHeyman]

Thanks. It's good to know that I am making mathematics easier/simpler.