I am reading Dummit and Foote Section 13.4 Splitting Fields and Algebraic Closures
In particular, I am trying to understand D&F's example on page 541 - namely "Splitting Field of a prime - see attached.
I follow the example down to the following statement:
" ... ... ... so the splitting field is precisely "
BUT ... then D&F write:
This field contains the cyclotomic field of roots of unity and is generated over it by , hence is an extension of at most p. It follows that the degree of this extension over is .
*** Can someone please explain the above statement and show formally and explicitly (presumably using D&F ch 13 Corollary 22 - see Note 1 below) why the degree of over is .
I also find it hard to follow the statement:
" ... ... ... Since both and are subfields, the degree of the extension over is divisible by p and p - 1. Since both these numbers are relatively prime, it follows that the extension degree is divisible by p(p-1) so that we must have
... ... "
*** Can someone please try to make the above clearer - why exactly is the degree of the extension over divisible by p and p - 1. What is the importance of "relatively prime" and why does equality hold in the statement regarding the degree of the extension?'
*** Finally, we are told that p is a prime, but where does the argument in the example depend on p being prime. ["Relatively prime" is mentioned in the context of p and p-1 but they are consecutive integers and hence are coprime anyway]
I would be grateful for some clarification of the above issues.
1. Corollary 22 (Dummit and Foote Section 13.2 Algebraic Extensions, page 529
Suppose that where m and n are relatively prime: (n, m) = 1.