Exercises for 4
[nal]
1
Exercises for 4
E4.1. Let F = cfw_XY XZ, X + YZ, YZ Z2 , X + Z2 . In each of the following situations, reduce
the polynomial X2 with respect to F:
(i) in Q[X, Y, Z] with respect to Lex with X
Y
(ii) in Q[X, Y, Z] with respect to De

Exercises for 5
[nal version]
1
Exercises for 5
Remark: a mock test where students can practise nding Gr bner bases has been set up
o
online on Blackboard.
E5.1. Recall that Buchbergers algorithm with respect to Lex with X
Y, applied to the
ideal <XY Y 1,

Exercises for 1
[nal version]
1
Exercises for 1
Exercises marked revision do not require any new material taught in this course (they are based on
what was covered in the prerequisites), and will typically not be discussed in class. Answers to most
exerci

5. Buchbergers algorithm
[revised v1 04/03/2014]
1
5. Buchbergers algorithm
We will now study an algorithm, due to Buchberger, for constructing a Gr bner basis of an
o
ideal I = <f1 , . . . , fk > of K[X]. We write K[X] to denote K[X1 , . . . , Xn ].
Deni

3. Ideals
[revised v1 11/02/2014]
1
3. Ideals
Reminder: ideal
Recall from Algebraic Structures 2 that an ideal of a ring R is a subset I of R such that:
0 I;
a, b I = a + b I;
a I, r R = ra I
that is, an ideal is a subset of R which is closed under ze