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simplifying expressions - How to Simplify equations over a Ring with Mathematica?


For example, when we work over a ring, the equation x^3=0 does not imply x^2=0 or x=0, but the vice versa is true. Can we use Mathematica to Simplify equations over a ring?



Answer



If you want to solve an equation over integer rings $\mathbb{Z}_n$ you should specify them with Modulus e.g.


Column[Solve[x^3 == 0, x, Modulus -> #] & /@ Range[2, 9]]

enter image description here


Edit


Since there was no further example of any expression to simplify over a finite ring let's define e.g. a polynomial which cannot be factorized over rationals (as Mathematica does by default)


p[x_] := x^5 + 3 x^4 + 6 x^3 - 2 x^2 + 1

Factor[p[x]]
p /@ Range[5]


1 - 2 x^2 + 6 x^3 + 3 x^4 + x^5
{9, 121, 631, 2145, 5701}

however over rings $\mathbb{Z}_n$ it is evaluated automatically with Mod[ p[x], n], (it has the Listable attribute), thus


Column[ Mod[p /@ Range[2, 10], #] & /@ Range[2, 10]]



 {{{1, 1, 1, 1, 1, 1, 1, 1, 1}},
  {{1, 1, 0, 1, 1, 0, 1, 1, 0}},
  {{1, 3, 1, 1, 1, 3, 1, 1, 1}},
  {{1, 1, 0, 1, 4, 1, 1, 0, 1}},
  {{1, 1, 3, 1, 1, 3, 1, 1, 3}},
  {{2, 1, 3, 3, 2, 1, 2, 2, 1}},
  {{1, 7, 1, 5, 1, 3, 1, 1, 1}},
  {{4, 1, 3, 4, 1, 6, 4, 1, 0}},
  {{1, 1, 5, 1, 9, 1, 1, 5, 1}} }


On the other hand you can use PolynomialMod to "simplify" a polynomial over a ring $\mathbb{Z}_n$, e.g.


Column[ PolynomialMod[ p[x], #] & /@ Range[2, 6] ]


1 + x^4 + x^5
1 + x^2 + x^5
1 + 2 x^2 + 2 x^3 + 3 x^4 + x^5
1 + 3 x^2 + x^3 + 3 x^4 + x^5
1 + 4 x^2 + 3 x^4 + x^5


So to get the table Column[ Mod[p /@ Range[2, 10], #] & /@ Range[2, 10]] as above, you can Apply as well PolynomialMod on a specific level of an adequate Table, e.g.


Column[ Apply[ PolynomialMod[ p[#2], #1] &, Table[{i, j}, {i, 2, 10}, {j, 2, 10}], {2}] ] === 
Column[ Mod[ p /@ Range[2, 10], #] & /@ Range[2, 10]]


True

In case you'd like to factorize p[x] over a finite field (for n prime $\mathbb{Z}_n$ is a field) it can be done with Modulus as well, e.g.


Column[ Factor[ p[x], Modulus -> #] & /@ Prime @ Range[4]]


enter image description here


Some related details (e.g. Extension to work with polynomials and algebraic functions over rings of Rationals extended by selected algebraic numbers) you could find here.


Consider another polynomial


w[x_] := 6 - 12 x + x^2 - 2 x^3 - x^4 + 2 x^5

you can solve the equation w[x] == 0 over the field of Rationals as well (by default Mathematica solves over Complexes, and then you needn't specify the domain), e.g.


Column[ Solve[w[x] == 0, x, #] & /@ {Integers, Rationals, Reals, Complexes} ]

enter image description here



You could factorize completely this polynomial with Extension :


Factor[ w[x]]
Factor[ w[x], Extension -> {Sqrt[2], Sqrt[3], I}]

enter image description here


There is also a package AbstractAlgebra to work with adequate algebraic concepts and a related book Exploring Abstract Algebra with Mathematica.


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