Kronecker-Weber via Ramification Theory Sharad V. Kanetkar In this note we prove the well known theorem of Kronecker-Weber using only ramification theory. The following steps are described in a series of exercises in [1, pp. 125-127]. Kronecker-Weber Theorem. Theorem : Every finite abelian extension of Q (field of rational numbers) is contained in a cyclotomic field. Proof : Let K be a finite abelian extension of Q with G = Gal(K/Q). Step 1 : It is enough to assume that K is of degree pm over Q for some prime p. For if G is expressed as a direct product of its Sylow subgroups : G∼ = Sp1 × · · · × Spr , then fixed subfields ki (of K) with groups Spi will generate K. If ki belongs to a cyclotomic field Fi , for i = 1, 2, · · · , r; then K ⊆ F1 F2 · · · Fr ⊆ some cyclotomic field. Hence we assume K = k1 and [K : Q] = pm . Step 2 : It is enough to assume that p is the only prime ramified in K. Suppose q ∈ Z is a prime (other that p) which is ramified in K. Let E(·|·) and e(·|·) denote the inertia group and the ramification index respectively. Let U be a prime of K lying above q with e(U | q) = e. Now the higher ramification group V1 (U | q) is a q−subgroup of a p−group G [1, page 121]. Hence |V1 (U | q)| = 1 and |V0 /V1 | = e. Since G is abelian |V0 /V1 | | (q − 1) [1, page 124, Ex. 26(c)]. This gives e|(q − 1). Now there is a (unique) subfield K1 ⊆ Q(ζq ) (where Q(ζm ) denotes the m-th cyclotomic field, i.e. ζm is a primitive m-th root of unity) with [K1 : Q] = e. Since e | pm and q 6= p, q is tamely ramified in both K1 and K. Now q is totally ramified in Q(ζq ) and hence in K1 . This gives that the ramification index of q in K1 is also e. Let U1 be a prime of L lying above U in K. Now, Gal(K/Q) and Gal(K1 /Q) are both p-groups and since Gal(L/Q) injects into Gal(K/Q)× Gal(K1 /Q), it is also a p-group. This shows that V1 (U1 | q) is both a p-group and a q-group implying that it is trivial. Thus E(U1 | q) is cyclic. Let W be the (unique) prime of K1 lying below U1 . Hence, by restriction, E(U1 | q) injects into E(U | q) × E(W | q). All these three groups are cyclic and the last two have 147 148 SHARAD V. KANETKAR order e each. This shows that E(U1 | q) is of order e. Thus the ramification index of q in L is also e. Since e(U1 | q) = e(U | q) = e, e(U1 | U ) = 1. Let L1 be the inertia field of U1 , i.e., L1 is the fixed field of E(U1 | q). Then for any field F containing L1 , U1 ∩ F is totally ramified in L. Thus for F = L1 K1 , (U1 ∩ F ) is totally ramified in L. But F ⊃ K1 and therefore e(U1 | (U1 ∩ F )) | e(U1 | U ). This implies e(U1 | U1 ∩ F ) = 1. Thus U1 ∩ F is totally ramified as well as unramified in L implying F = L. Hence if L1 belongs to a cyclotomic field then since K1 ⊂ Q(ζq ), L will be a subfield of a cyclotomic field. But K ⊂ L and hence K will be a subfield of some cyclotomic field proving the theorem. Thus it is enough to replace K by L1 . But it is easy to see that all unramified primes of K are unramified in L1 and, in addition, q is also unramified in L1 (but ramified in K). Thus continuing this process of reduction we can assume that there are no primes other that p which are ramified in K. This finishes the proof of step 2. Step 3 : Case(i) p = 2, [K : Q] = 2m . In this case 2 is totally ramified in K since otherwise no prime will be ramified in the fixed field of E(U | 2) and this will imply, by [1,page 137, Cor.3], that E(U | 2) = G. Thus 2 is totally √ ramified in K. Thus e(U | 2) = 2m . If m = 1 then [K : Q] = 2 and K = Q[ d] for some square-free integer d. But the Disc(K/Q) = d or 4d. Since 2 is the only ramified prime of K, 2 is the only possible divisor of d. Hence √ √ √ K = Q[ 2] or Q[ −2] or Q[ 1]. All these fields are subfields of Q[ζ8 ]. Hence the theorem is proved in this case. If m > 1 then consider L = Q(ζ2m+2 ) ∩ R, where R is the field of real numbers. Then [L : Q] = 2√m and L ⊂ R. Hence L contains a unique quadratic subfield, namely Q[ 2]. Hence Gal(L/Q) contains unique subgroup of index 2. Thus L is a cyclic extension. Now consider the field LK. Let µ be the extension of σ (where < σ >= Gal(L | Q) to LK. Let F be the fixed field of µ. Since µ restricted to L generates√Gal(L/Q), F ∩ L = Q. If [F : Q] > 2 then F ∩R 6= Q and it will contain√ Q[ 2] ⊂ L but F ∩L = Q. Hence [F : Q] ≤ 2. If [F : Q] = 2 then F = Q[ −2] or Q[i] and both are contained in Q[ζ8 ]. Thus K ⊆ LK = F L ⊆ Q(ζ2m+2 ) and the theorem is proved. If F = Q then < µ >= Gal(LK/Q) and since Gal(LK/Q) ,→ Gal(L/Q) × Gal(K/Q), order of any element of Gal(LK/Q) ≤ lcm (| Gal (L/Q) |, | Gal (K/Q) |) = 2m . Thus 2m ≤ [LK : Q] ≤ 2m . Hence L = LK implying K ⊆ L ⊆ Q[ζ2m+2 ]. Thus the theorem is proved in this case also. KRONECKER - WEBER THEOREM 149 Case(ii) p is odd and [K : Q] = pm . Consider the case m = 1. Hence K is of degree p over Q and p is the only ramified prime in K. Thus if U is the prime of K lying above p then e(U | p) = p. Claim : diff(R/Z) = U 2(p−2) , where R is the ring of integers of K. Proof : Let π ∈ U − U 2 then π satisfies a monic irreducible polynomial over Z, say, f (x) = xp + ap−1 xp−1 + · · · + a0 . Let ϑU be the valuation corresponding to the DVR RU . Then ϑU (π) = 1 and since U p = pR, ϑU (p) = p. Now the coefficients ai are symmetric polynomials in σπ, σ ∈ Gal(K/Q) and ϑU (σπ) = 1, ∀σ ∈ Gal(K/Q). Hence Q ϑU (ai ) ≥ 1 and hence p | ai . But a0 = ± (σπ) and hence ϑU (a0 ) = p. Now in the expression f 0 (π) = pπ p−1 + (p − 1)ap−1 π p−2 + · · · + a1 , all terms have valuations distinct mod p. Therefore ϑU (f 0 (π)) = min{ϑU (pπ p−1 ), ϑU ((p − 1)ap−1 π p−2 · · · ϑU (a1 )}. Hence, 2p − 1 ≥ ϑU (f 0 (π)) ≥ p. But by Hilbert’s formula [1, page 124, Exc. 27], ϑU (f 0 (π)) = ϑU (diff(R/Z)) = ∞ X (| Vi | −1) i=0 Since | Vi | is a power of p, (p − 1) | ϑU (f 0 (π)). Hence ϑU (f 0 (π)) = 2p − 2. And diff(R/Z) = U 2p−2 (because no other prime is ramified in k). Thus the claim is proved. Now let m = 2. Claim : G is cyclic. Proof : Consider the inertia field corresponding to the prime p. In this field p is unramified. Hence no prime is ramified in this inertia field. Hence it must be equal to Q. Thus K is totally ramified with e(U/p) = p2 . Since V1 is Sylow−p subgroup of Gal(K/Q), | V1 |= p2 =| V0 | . Let Vr = Vr (U/p) be the least r for which | Vr |< p2 . But Vr−1 /Vr ,→ R/U ∼ = Z/pZ and hence | Vr |= p. Let H be any subgroup of G having order p. Let KH be the fixed field of H. Then [KH : K] = p and diff(RH /Q]) = U 2p−2 . Hence from the transitivity of different, diffR/Z) = diff(R/RH ).U (2p−2)p , [1, page 96, Ex.38]. 150 SHARAD V. KANETKAR Hence diff(R/RH ) is independent of H as long as [H : Q] = p. Now by Hilbert’s formula the power of U dividing diff(R/RH ) is given by α= ∞ X | Vi ∩ H | −1. i=0 Hence α is strictly maximized when H = Vr . Since α is independent of H, Vr is the only subgroup of order p in G. Thus G is cyclic, proving the claim. Thus in case m = 1, k is unique, otherwise KK1 will be of degree p2 containing two distinct subfields of degree p. Hence K is the unique subfield of Q[ζp2 ]. Thus the theorem is true for the case m = 1. Now let m > 1. Let L denote the unique subfield of Q[ζpm+1 ] of degree pm over Q. Then Gal(L/Q) is cyclic of order pm . Then LK is cyclic by the claim. But Gal(LK/Q) ,→ Gal(L/Q) × Gal(k/Q), hence, | Gal(LK/Q) | ≤ lcm(| Gal(L/Q) |, | Gal(K/Q) |) = pm . Therefore L ⊆ LK ⊆ L and hence K ⊆ L ⊆ Q(ζpm+1 ), and the theorem is proved in this case also. REFERENCE 1. D. A. Marcus, Number Fields, Springer Verlag, 1977. Sharad V. Kanetkar Bhaskaracharya Pratishthana 56/14, Erandavane, Damle Path Off Law College Road Pune-411 004 e-mail : bhaskara− [email protected]

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