DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1. If every e equals 1, we say p s unramfed n K. Example 1.1. In Z[], the only prme whch ramfes s 2: (2) = (1 + ) 2. Example 1.2. Let K = Q(α) where α s a root of f(x) = T 3 9T 6. Then 6 = α 3 9α = α(α 3)(α + 3). For m Z, α + m has mnmal polynomal f(t m) n Q[T ], so N K/Q (α + m) = f( m) = m 3 9m + 6 and the prncpal deal (α m) has norm N(α m) = m 3 9m + 6. Therefore N(α) = 6, N(α 3) = 6, and N(α + 3) = 6. It follows that (α) = p 2 p 3, (α 3) = p 2 p 3, and (α + 3) = p 2 p 3 (so, n partcular, α + 3 and α 3 are unt multples of each other). Thus (2)(3) = (6) = (α)(α 3)(α + 3) = p 2 p 2 2 p 3 3, so (2) = p 2 2 p 2 and (3) = p3 3. Ths shows 2 and 3 are ramfed n K. Note that one of the exponents n the factorzaton of (2) exceeds 1, whle the other equals 1. One way to thnk about ramfed prmes s n terms of the rng structure of O K /(p). By (1.1) and the Chnese Remander Theorem, (1.2) O K /(p) = O K /p e 1 1 O K/p eg g. If some e s greater than 1, then the quotent rng O K /p e has a nonzero nlpotent element (use the reducton modulo p e of any element of p p e ), so the product rng (1.2) has a nonzero nlpotent element. If each e equals 1, then O K /(p) s a product of (fnte) felds, and a product of felds has no nonzero nlpotent elements. Thus, p ramfes n K f and only f O K /(p) has a nonzero nlpotent element. Our goal n ths handout s to prove the followng result, whch characterzes the prme numbers ramfyng n a number feld n terms of the dscrmnant. Theorem 1.3. For a number feld K, the prmes whch ramfy are those dvdng the nteger dsc Z (O K ). Snce dsc Z (O K ) 0, only fntely many prmes ramfy n K. 1
2 KEITH CONRAD 2. The power-bass case We wll frst consder Theorem 1.3 when O K has a power bass over Z. The treatment of the general case n Secton 3 wll not rely on the power-bass case at all, but the power-bass case s techncally smpler. We wll just sketch the basc deas behnd ths specal case. Proof. Assume O K = Z[α] for some α O K. Let f(t ) be the mnmal polynomal of α over Q, so f(t ) s monc n Z[T ]. Wth po K factorng nto prmes as n (1.1), by defnton p ramfes n O K f and only f some e > 1. The factorzaton of the mod p reducton f(t ) n (Z/pZ)[T ] s f(t ) = π e 1 1 πeg g for some dstnct monc rreducbles π (Z/pZ)[T ]. The π (T ) s are separable (all rreducbles over a fnte feld are separable), so some e > 1 f and only f f(t ) has a repeated root n a splttng feld over Z/pZ. Ths s equvalent to f(t ) havng dscrmnant 0: p ramfes n O K f and only f dsc(f) = 0 n Z/pZ. Snce the dscrmnant of a monc polynomal s a unversal polynomal n ts coeffcents (consder the quadratc case, where T 2 + bt + c has dscrmnant b 2 4c), dscrmnants of monc polynomals behave well under reducton: dsc(f(t ) mod p) = dsc(f(t )) mod p. Therefore dsc(f(t )) = 0 n Z/pZ f and only f dsc(f) 0 mod p. Thus p ramfes n O K f and only f p dsc(f). Snce O K = Z[α] = Z[T ]/(f(t )), dsc(f) = dsc Z (Z[T ]/(f)) = dsc Z (O K ). 3. The general case To prove Theorem 1.3 for every O K, we wll examne dscrmnants of rng extensons to show computng the dscrmnant commutes wth reducton mod p: dsc Z (O K ) mod p = dsc Z/pZ (O K /(p)). Then we wll use (1.2) to wrte dsc Z/pZ (O K /(p)) as a product of dscrmnants of rngs of type O K /p e and compute the dscrmnants of these partcular rngs. Defnton 3.1. Let A be a commutatve rng and B be a rng extenson of A whch s a fnte free A-module: B = Ae 1 Ae n. Then we set dsc A (e 1,..., e n ) = det(tr B/A (e e j )) A. Remark 3.2. The dscrmnant of a bass s an algebrac concept of volume. To explan ths vewpont, we should thnk about Tr B/A (xy) as an analogue of the dot product v w n R n. For a bass v 1,..., v n n R n, the ordnary Eucldean volume of the parallelotope { } a v : 0 a 1 havng edges v s det(v v j ). The dscrmnant of an A-bass of B uses the A-valued parng x, y = Tr B/A (xy) on B n place of the R-valued dot product on R n and we just drop the absolute value and the square root when we make the algebrac analogue. How are the dscrmnants of two A-module bases for B related? Pck a second bass e 1,..., e n of B as an A-module. Then e = a j e j, j=1
DISCRIMINANTS AND RAMIFIED PRIMES 3 where a j A and the change of bass matrx (a j ) has determnant n A. Then ( ) Tr B/A (e e j) = Tr B/A a k e k a jl e l k=1 so Therefore We set = k=1 l=1 l=1 a k Tr B/A (e k e l )a jl, (Tr B/A (e e j)) = (a j )(Tr B/A (e e j ))(a j ). dsc A (e 1,..., e n) = (det(a j )) 2 dsc A (e 1,..., e n ). dsc A (B) = dsc A (e 1,..., e n ) A for any A-module bass {e 1,..., e n } of B. It s well-defned up to a unt square. In partcular, the condton dsc A (B) = 0 s ndependent of the choce of bass. Gven a number feld K, ramfcaton of the prme p n K has been lnked to the structure of the rng O K /(p) n Secton 1. Let s look at the dscrmnant of ths rng over Z/pZ. Lettng K have degree n over Q, the rng O K s a free rank-n Z-module, say O K = Zω. Reducng both sdes modulo p, O K /(p) = (Z/pZ)ω, so O K /(p) s a vector space over Z/pZ of dmenson n. The dscrmnant of O K s dsc Z (O K ). The next lemma says reducton modulo p commutes (n a sutable sense) wth the formaton of dscrmnants. Lemma 3.3. Choosng bases approprately for O K and O K /(p), dsc Z (O K ) mod p = dsc Z/pZ (O K /(p)). Proof. Pck a Z-bass ω 1,..., ω n of O K. The reductons ω 1,..., ω n n O K /(p) are a Z/pZbass, so the multplcaton matrx [m x ] for any x O K, wth respect to the bass {ω }, reduces modulo p to the multplcaton matrx [m x ] for x on O K /(p) wth respect to the bass {ω }. Therefore Tr (OK /(p))/(z/pz)(x) = Tr(m x ) = Tr(m x ) mod p = Tr OK /Z(x) mod p. Thus, the mod p reducton of the matrx (Tr OK /Z(ω ω j )) s (Tr (OK /(p))/(z/pz)(ω ω j )). Now take determnants. Lemma 3.4. Let A be a commutatve rng and B 1 and B 2 be commutatve rng extensons of A whch are each fnte free A-modules. Then, choosng A-module bases approprately, dsc A (B 1 B 2 ) = dsc A (B 1 ) dsc A (B 2 ).
4 KEITH CONRAD Proof. Pck A-module bases for B 1 and B 2 : m B 1 = Ae, B 2 = Af j. As an A-module bass for B 1 B 2 we wll use the m + n elements e 1,..., e m, f 1,..., f n. Snce e f j = 0 n B 1 B 2, the matrx whose determnant s dsc A (B 1 B 2 ) s a block dagonal matrx ( ) (Tr(B1 B 2 )/A(e e k )) O. O (Tr (B1 B 2 )/A(f j f l )) For any x B 1, multplcaton by x on B 1 B 2 klls the B 2 component and acts on the B 1 -component n the way x multples on B 1, so a matrx for multplcaton by x on B 1 B 2 s a matrx whose upper left block s a matrx for multplcaton by x on B 1 and other blocks are 0. Thus Tr (B1 B 2 )/A(x) = Tr B1 /A(x) for x B 1. Smlarly, Tr (B1 B 2 )/A(x) = Tr B2 /A(x) for x B 2. Thus ( ) ( (Tr(B1 B 2 )/A(e e k )) O (TrB1 = /A(e e k )) O O (Tr (B1 B 2 )/A(f j f l )) O (Tr B2 /A(f j f l )) and takng determnants gves Now we prove Theorem 1.3. j=1 dsc A (B 1 B 2 ) = dsc A (B 1 ) dsc A (B 2 ). Proof. We have p dsc Z (O K ) f and only f dsc Z (O K ) 0 mod p. By Lemma 3.3 dsc Z (O K ) mod p = dsc Z/pZ (O K /(p)), so p dsc Z (O K ) f and only f dsc Z/pZ (O K /(p)) = 0 n Z/pZ. In (1.2), each factor O K /p e s a Z/pZ-vector space snce p p e. Usng (1.2) and Lemma 3.4, g dsc Z/pZ (O K /(p)) = dsc Z/pZ (O K /p e ). Therefore we need to show for any prme number p and prme-power deal p e such that p e (p) that dsc Z/pZ (O K /p e ) s 0 n Z/pZ f and only f e > 1. (Recall that the vanshng of a dscrmnant s ndependent of the choce of bass.) Suppose e > 1. Then any x p p e s a nonzero nlpotent element n O K /p e. By lnear algebra over felds, such an x can be used as part of a Z/pZ-bass of O K /p e, say {x 1,..., x n } wth x = x 1. Wrtng the trace map Tr (OK /p e )/(Z/pZ) as Tr for short, the frst column of the matrx (Tr(x x j )) contans the numbers Tr(x x). These traces are all 0: x x s nlpotent, so the lnear transformaton m x x on O K /p e s nlpotent and thus ts egenvalues all equal zero. Snce one column of the trace-parng matrx (Tr(x x j )) s all 0, dsc Z/pZ (O K /p e ) = 0. Now suppose e = 1. Then O K /p e = O K /p s a fnte feld of characterstc p. We want to prove dsc Z/pZ (O K /p) 0. If ths dscrmnant s 0, then (because O K /p s a feld) the trace functon Tr: O K /p Z/pZ s dentcally zero. However, from the theory of fnte felds, ths trace functon can be wrtten as a polynomal functon: Tr(t) = t + t p + t p2 + + t pr 1, ),
DISCRIMINANTS AND RAMIFIED PRIMES 5 where #(O K /p) = p r. Snce the degree of ths polynomal s less than the sze of O K /p, ths functon s not dentcally zero on O K /p. Therefore the dscrmnant of a fnte extenson of Z/pZ does not equal zero.