4

SHREERAM S. ABHYANKAR AND MANISH KUMAR

R' of R we have that the integral closure S' of R' in L is a two dimensional regular

semilocal domain.

(1.2) Assume that

n

= 1 and there exists 0-=/= o: E K together with (3 E K such

that for

F{

=

F{ (

Z) = o:-

2

F1 ( o:Z

+

(3) = Z

2

+

A' Z

+

B'

we have

F{(Z)

E R[Z] with ordRF{(Z) = 1. Then Sis two dimensional regular

semilocal domain.

(1.3) Assume that char(K) = 2 = n and

A1 = xya and A2 = XYaE with B1 =

B2

= Y

where

a

= a positive integer

and

E

= xtyu with nonnegative integers

t, u

at least one of which is positive.

Let H be a root of F

1

in

£

1. Then 8

1

is a two dimensional regular local domain

with M(Sl) = (X, H)S1

,

and S is a two dimensional nonregular local domain.

Moreover,

L/ K

is Galois with Galois group Z

2

EB Z

2

.

PROOF.

To prove

(1.0)

note that L1 = K(H1

).

Also note that (i) follows from

the discriminant theory given in [A03], (i') and (i") are straightforward, and, in

view of (i') (resp: (i")), (ii') (resp: (ii")) follows by taking

{

(Rj, Lj, X, Hi) and (F1, ... , F1-1, FJ+l, ... , Fn)

(resp: (Rz, Lz, Hz, Y) and (F1, ... , Fz-1, Fl+l, ... , Fn))

for (R, K, X, Y) and (F1, ... , Fn) in (i). Likewise, in view of (i'), (iii) follows by

taking

(RJ, Li, X, Hi) and (F1, ... , F1-1, Fi+l' ... , Fn)

for (R, K, X, Y) and (F1, ... , Fn) in (i"). Likewise, in view of (i') (resp: (i")), (iii')

(resp: (iii")) follows by taking

{

(Rj,LJ,X,HJ) and (F1, ... ,Fj-l,FJ+l,···,Fn)

(resp: (Rz,Lz,Hz,Y) and (Fl, ... ,Fz-1,Fl+l,···,Fn))

for (R,K,X,Y) and (F1

, ...

,Fn) in (ii") (resp: (ii')). (iv) follows from (ii'), (ii"),

(iii), (iii'), and (iii"). To prove (v) suppose that

(J'

U J") =

0

and J

111

-=/=

0

and let

R'

is any two dimensional quadratic transform of

R.

Suitably relabelling

X, Y

we

may assume that Y/X E R'. Now ifY/X E M(R') then M(R') =(X, Y/X)R' and

for all j E J we have

C1

=

D1Xrjysj

where

D

1

E R' \ M(R') with

rj

=

r1 + s1

and

sj

=

s1,

and hence

rj

= 0 with

sj

= 0 or 1, and so we are reduced to (iv). Likewise if

Y /X

'f.

M(R') then

M(R') = (X, Y')R' for some Y' E M(R'), and for all j E J we have

C1

=

Djxri

(Y')sj

where

Dj

=

D

1

(Y/X)

8 J

E R' \ M(R') with

rj

=

r1 + s1

with

sj

= 0, and hence

rj + sj

= 0, and so we are again reduced to (iv).

(1.1)

follows from parts (iv) and (v) of

(1.0).