### Kronecker-Weber via Ramification Theory

```Kronecker-Weber via Ramification Theory
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
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
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.
56/14, Erandavane, Damle Path