- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey!
Let b a prime, and let be irreducible. Let be a root pf .
We have that , where is a basis of , so . There are choices for each . So, there are choices for . Therefore, we have that , right? (Wondering)
I want to show that contains all the roots of .
I holds that every finite extension of a finite field is normal, right? But how can we prove this? (Wondering)
Then I have shown that for each , is a root of . I have also shown that and for some .
Then I want to show that is cyclic and let be a generator. I want to calculate also the order of as a function of .
In the book there is the following corollary:
View attachment 6352
We have that and . From the above we have that is Galois, , which is true since . From the theorem we also have that is cyclic and is generated by the automorphism .
How can we calculate the order of ? (Wondering)
Then independent from that I want to calculate a simple expression of .
I have done the following:
Since is a generator of that means that is a -automorphism of that maps to a root of to an other root, right? (Wondering)
So, we have that . Therefore, we get the following:
Is this correct? (Wondering)
The order of is the smallest integer such that . Does this imply that ? (Wondering)
Let
We have that
I want to show that
I holds that every finite extension of a finite field is normal, right? But how can we prove this? (Wondering)
Then I have shown that for each
Then I want to show that
In the book there is the following corollary:
View attachment 6352
We have that
How can we calculate the order of
Then independent from that I want to calculate a simple expression of
I have done the following:
Since
So, we have that
The order of