12   Different and discriminant  Notation  Let x 1       x n   L  Set         x 1       x n     det   Tr L   K   x i x j       K   det     k   1 n   k   x i     k   x j       det   B B         where   k   L   K   are distinct embeddings and B       i   x j       Note   If y i     j   1 n a i j x j  a i j   K  then          y 1       y n     det   A   2     x 1       x n       where A     a i j     If  x 1       x n   O L  then       x 1       x n     O K   Lemma 12 1  Assuming that   k a perfect field  R a k algebra which is finite dimensional as a k vector space  Then  the Trace form                 R   R   R   x   y     Tr R   k   x y         Tr k   mult   x y           is non degenerate if and only if R   k 1       k r where k i   k is a finite separable extension of k   Proof  Example Sheet 3     Theorem 12 2  Assuming that   0       O K prime ideal  Then    i  If   ramifies in L  then for every  x 1       x n   O L  we have      x 1       x n     0 mod      ii  If   is unramified in L  then there exists x 1       x n such that            x 1       x n       Proof   i  Let    O L   P 1 e 1   P r e r   0   P i   O L distinct prime ideals  e i   0  Define      R     O L     O L   CRT   i   1 r O L   P i e i       If   ramifies  then   O L     O L has nilpotents  Hence          x   1       x   n     0   x   i   O L     O L       Then using the fact that   commutes  we get that          x 1       x n     0 mod     x i   O L     O L        ii    unramified implies  R   O L     O L is a product of finite extensions of k  By Lemma 12 1  we get that the Trace form is non degenerate  hence for x   1       x   n a basis of   O L     O L as a k vector space  we have      x   1       x   n     0  So thee exist  x 1       x n   O L such that          x 1       x n       0 mod           Definition 12 3   Discriminant    The discriminant  is the ideal  d L   K   O K generated by      x 1       x n   for all choices of  x 1       x n   O L   Corollary 12 4    ramifies L if and only if      d L   K  In particular  only finitely many primes ramify in L   Definition 12 5   Inverse different    The inverse different  is      D L   K   1     y   L   Tr L   K   x y     O K   x   O L         an  O L submodule of L   Lemma 12 6  D L   K   1 is a  fractional ideal in L   Proof  Let  x 1       x n   O L a K basis for L   K  Set     d         x 1       x n     det   Tr L   K   x i x j           which is non zero since separable   For  x   D L   K   1 write x     j   1 r   j x j with   j   K  We show    j   1 d O K  We have      Tr L   K   x x i       j   1 n   j Tr L   K   x i x j     O K       Set A i j   Tr L   K   x i x j    Multiplying by  Adj   A     M n   O K    we get      d     1     n     Adj   A     Tr L   K   x x 1     Tr L   K   x x n         Since    i   1 d O K  we have  x   1 d O L  Thus   D L   K   1   1 d O K  so  D L   K   1 is a fractional ideal     The inverse D L   K of  D L   K   1 is the different ideal   Remark   D L   K   O L since   O L   D L   K   1   Let I L  I K be the groups of fractional ideals   Theorem 9 7 gives that      I L     0   P prime ideals in O L     I K     0   P prime ideals in O K       Define N L   K   I L   I K induced by P     f for      P   O K and f   f   P         Fact     Use Corollary 10 10 and   v     N L P   K     x       f P     v   x   for msubx   msub   where  v   and  v P are the normalised valuations for  L P   K      Theorem 12 7  N L   K   D L   K     d L   K   Proof  First assume  O K   O L are PIDs  Let x 1       x n be an  O K basis for  O L and y 1       y n be the dual basis with respect to trace form  Then y 1       y n is a basis for  D L   K   1  Let   1         n   L   K   be the distinct embeddings  Have        i   1 n   i   x j     i   y k     Tr   x j y k       j k       But          x 1       x n     det     i   x j     2       Thus          x 1       x n       y 1       y n     1       Write   D L   K   1     O L since     L  Then     d L   K   1         x 1       x n     1           y 1       y n               x 1         x n     change of basis matrix is invertible in O K   N L   K     2       x 1       x n   change of basis matrix is   mult             Thus      d L   K   1   N L   K   D L   K   1   2 d L   K     so      N L   K   D L   K     d L   K       In general  localise at  S   O K     and use   S   1 D L   K   D S   1 O L   S   1 O K  Then   S   1 d L   K   d S   1 O L   S   1 O K  Details omitted     Theorem 12 8  Assuming that   O L   O K          has monic minimal polynomial  g   X     O K   X    Then   D L   K     g             Proof  Let       1         n be the roots of g  Write      g   X   X         n   1 X n   1         1 X     0     with    i   O L and   n   1   1  We claim        i   1 n g   X   X     i   i n g       i     X r     for 0   r   n   1   Indeed the difference is a palynomial of degree   n  which vanishes for X     1         n  Equate coefficients of X s  which gives      Tr L   K     r   s g               r s       Since 1             n   1 is an  O K basis for  O L   D L   K   1 has an  O K basis       0 g             1 g                 n   1 g           1 g               Note all of these are  O L multiples of the last term  since the   i are in  O L  So  D L   K   1   1   g            hence  D L   K     g               P a prime ideal of  O L       O K   P   D L P   K   using  O K     O L P  We identify  D L P   K   with a power P   Theorem 12 9  D L   K     P D L P   K    finite product  see later     Proof  Let x   L       O K  Then     Tr L   K   x       P     Tr L P   K     x            of Corollary 10 10     Let   r   P     v P   D L   K     s   P     v P   D L P   K            i e  r   P     s   P     Let x   L with  v P   x       s   P   for all P  Then  Tr L P   K     x y     O K    for all y   L and for all P  Using      we get      Tr L   K   x y     O K     y   O L     P       Thus      Tr L   K   x y     O K   y   O L     so   D L   K     P D L P   K          i e  r   P     s   P     Fix P and let x   P   r   P     P   r   P     1  Then  v P   x       r   P     v P     x     0 for all P     P  By       we have      Tr L P   K     x y     Tr L   K   x y       P       P     P Tr L P   K     x y     y   O L     hence      Tr L P   K     x y     O K     y   O L P       Hence msubsup  x   msubsup   K     1  i e     v P   x     r   P     s   P    So   D L   K     P D L P   K       Corollary 12 10  d L   K     P     d L P   K     Proof  Apply  N L   K to   D L   K     P     D L P   K