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- Jun 22, 2012

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In particular, I am trying to understand D&F's example on page 541 - namely "Splitting Field of [TEX] x^p - 2, p [/TEX] a prime - see attached.

I follow the example down to the following statement:

" ... ... ... so the splitting field is precisely [TEX] \mathbb{Q} ( \sqrt[p]{2}, \zeta_p ) [/TEX]"

BUT ... then D&F write:

This field contains the cyclotomic field of [TEX] p^{th} [/TEX] roots of unity and is generated over it by [TEX] \sqrt[p]{2} [/TEX], hence is an extension of at most p. It follows that the degree of this extension over [TEX] \mathbb{Q} [/TEX] is [TEX] \le p(p-1) [/TEX].

*** 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 [TEX] \mathbb{Q} ( \sqrt[p]{2}, \zeta_p )[/TEX] over [TEX] \mathbb{Q} [/TEX] is [TEX] \le p(p-1) [/TEX].

I also find it hard to follow the statement:

" ... ... ... Since both [TEX] \mathbb{Q} ( \sqrt[p]{2} ) [/TEX] and [TEX] \mathbb{Q} ( \zeta_p ) [/TEX] are subfields, the degree of the extension over [TEX] \mathbb{Q} [/TEX] 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

[TEX] [\mathbb{Q} ( \sqrt[p]{2}, \zeta_p ) \ : \ \mathbb{Q}] = p(p - 1) [/TEX] ... ... "

*** Can someone please try to make the above clearer - why exactly is the degree of the extension over [TEX] \mathbb{Q} [/TEX] 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.

Peter

Note

1. Corollary 22 (Dummit and Foote Section 13.2 Algebraic Extensions, page 529

Suppose that [TEX] [K_1 \ : \ F] = n, [ K_2 \ : \ F ] = m [/TEX] where m and n are relatively prime: (n, m) = 1.

Then [TEX] [K_1K_2 \ : \ F] = [K_1 \ : \ F] [ K_2 \ : \ F ] = nm [/TEX]

2. The above has also been posted on MHF