Use Macaulay2 to show that a quotient ring is a field

Question

I am new to Macaulay2. I tried to use the following command to show that a quotient ring S=ZZ_977[x]/<7x^11+4x^5-23x^4+x-27> is a field:

i2 : S = ZZ[x]/<7*x^11+4*x^5-23*x^4+x-27>
o2 = S
     IsField S

But it doesn't work. I looked it up on internet but they don't cover this part. Can someone help? Thanks!

Answer 1

The command (or method in M2 speak) you're looking for is isField rather than IsField.

i1 : R = ZZ[x];

i2 : I = ideal(7*x^11+4*x^5-23*x^4+x-27);

o2 : Ideal of R

i3 : S = R/I;

i4 : isField S

o4 = false

i5 : isField (ZZ/2)

o5 = true

But there is (or at least was as of February 2014) a major caveat about using this internal M2 method for testing whether a quotient ring is a field (alternately testing the maximality of the ideal).

More generally, the applicable rules from the Macaulay2 style guide for naming methods and variables are

Names representing methods must be verbs and written in mixed case starting with lower case.

The prefix "is" should be used for Boolean variables and methods.