Some progress on Macaulay2 skills

Worked on MT4517 exercises chapter 2.

Show that { a + bX + cX^2 | a,b,c in Z2 and X^3 = 1 + X } is a field and calculate inverses of X, X^2 and 1+X.

Simple.

i96 : describe(R)

ZZ
-- [X]
2
o96 = ----------
3
X  + X + 1


i97 : ( (X)_R )^-1

2
o97 = X  + 1

o97 : R


i98 : ( (X^2) )^-1

2
o98 = X  + X + 1

o98 : R


i99 : ( (1+X)_R )^-1

2
o99 = X  + X

o99 : R

It would be cool to be able to do something like asList(R) where Macaulay2 should reply with 0, 1, 1+X, 1+X^2, etc.

For now I'll do it like this.

i136 : i=0; while i < 7 list (R_0)^i do i = i+1

2          2       2           2
o137 = {1, X, X , X + 1, X  + X, X  + X + 1, X  + 1}

o137 : List

Macaulay 2

Today I installed Macaulay 2, 'software system devoted to supporting research in algebraic geometry and commutative algebra'.

Example 1: calculate (x+y)^2 in the polynomial ring Z[x,y].

i107 : R=ZZ[x,y]
o107 = R
o107 : PolynomialRing
i108 : (x+y)^2
        2           2
o108 = x  + 2x*y + y
o108 : R

Example 2: calculate (x+y)^2 in the polynomial ring Z/2[x,y].

i111 : R=ZZ/2[x,y]
o111 = R
o111 : PolynomialRing
i112 : (x+y)^2
        2    2
o112 = x  + y
o112 : R