Derivation of the Kummer Class Formula: Step One

In the next series of blogs, I will show how the Kummer Class Formula is derived from the zeta function for cyclotomic integers. In a previous blog, I showed Kummer's proof that Fermat's Last Theorem was true for regular primes. The class formula is the method for determining whether a given prime is regular or not.

The content in today's blog is taken from Harold M. Edwards Fermat's Last Theorem: A Genetic Introduction

Lemma 1:

If β is an fth root of unity and n=ef, then:

(x - yβ)*...*(x - yβⁿ) = (x^f - y^f)^e

Proof:

(1) X^f - 1 has f distinct roots by the Fundamental Theorem of Algebra (see here)

(2) This means that:

X^f - 1 = (X - 1)(X - β)(X - β²)*...*(X - β^f-1)

(3) Since β^f = 1, we have:

X^f - 1 = (X - β)(X - β²)*...*(X - β^f-1)(X - β^f)

(4) Let X = x/y, then we have:

(x/y)^f - 1 = ([x/y] - β)([x/y] - β²)*...*([x/y] - β^f)

(5) If we multiply both sides by y^f, we get:

x^f - y^f = (x - βy)(x - β²y)*...*(x - β^fy)

(6) Further, if we put both sides to the power of e, we get:

(x^f - y^f)^e = [(x - βy)(x - β²y)*...*(x - β^fy)]^e =

= (x - βy)(x - β²y)*...*(x - β^fy)*
(x - β^f+1y)(x - β^f+2)*...*(x - β^2fy)*
...
(x - β^f(e-1)+1)(x - β^f(e-1)+2)*...*(x - β^efy) =

= (x - βy)*x - β²y)*...*(βⁿy)

QED

Theorem 2:



Proof:

(1) Since N(P)=p^f and N(α - 1)=λ (see here for p^f and here for N(α-1)=λ) and since there are e prime divisors P that divide a prime p (see here), we have:



(2) For each p, we can see that:

(1 - p^-fs)^-e = (1^f - (p^-s)^f)^-e

(3) Let γ be a primitive root mod λ

(4) For each p, let γ^j ≡ p (mod λ)

(5) Let β^j be an fth root of unity such that (β^j)^f = 1 but for any integer i less than f, (β^j)^f ≠ 1.

(6) Using Lemma 1 above, we have:

(1^f - (p^-s)^f)^e = (1 - β^jp^-s)*...*(1 - p^-sβ^jn) = ∏ (k=1, λ-1) (1 - β^jkp^-s) =

=∏ (k=0, λ-2) (1 - β^jkp^-s)

(7) Using step #6, we have:

(1 - λ^-s)^-1 ∏ (p ≠ λ) (1 - p^-fs)^-e = (1 - λ^-s)^-1 ∏ (p ≠ λ) ∏ (k=0, λ-2) (1 - p^-sβ^jk)^-1

(8) From this step, the conclusion follows:



QED

References

• Harold M. Edwards, Fermat's Last Theorem: A Genetic Introduction, Springer, 2000