Define a Field in Mathematica

Defining a Field in Mathematica

Mathematica is a powerful computational software system used in various fields of science, engineering, and mathematics. It provides a robust platform for symbolic and numerical computations, data visualization, and programming. In Mathematica, a field is a mathematical construct that is used to define a set of elements with specific properties and operations. In this post, we will discuss how to define a field in Mathematica.

What is a Field?

A field is a set of elements with two binary operations, addition and multiplication, that satisfy certain properties. The elements of a field are called scalars, and the operations are called addition and multiplication. The properties that a field must satisfy are:

• Commutativity of Addition: The order of the elements being added does not matter, i.e., a + b = b + a.
• Associativity of Addition: The order in which we add three or more elements does not matter, i.e., (a + b) + c = a + (b + c).
• Existence of Additive Identity: There exists an element 0 such that a + 0 = a.
• Existence of Additive Inverse: For each element a, there exists an element -a such that a + (-a) = 0.
• Commutativity of Multiplication: The order of the elements being multiplied does not matter, i.e., a * b = b * a.
• Associativity of Multiplication: The order in which we multiply three or more elements does not matter, i.e., (a * b) * c = a * (b * c).
• Existence of Multiplicative Identity: There exists an element 1 such that a * 1 = a.
• Existence of Multiplicative Inverse: For each non-zero element a, there exists an element 1/a such that a * (1/a) = 1.

Defining a Field in Mathematica

To define a field in Mathematica, we can use the Field function from the GAP package. However, Mathematica does not have a built-in Field function. We can define a field manually by creating a set of elements and defining the addition and multiplication operations.

For example, let’s define a field of integers modulo 5.

IntegersMod5 = Range[5];

AdditionMod5[a_, b_] := Mod[a + b, 5];

MultiplicationMod5[a_, b_] := Mod[a*b, 5];

In this example, we define the set of integers modulo 5 as IntegersMod5. We then define the addition and multiplication operations modulo 5 using the Mod function.

Properties of a Field

We can verify that the field we defined satisfies the properties of a field.

(* Commutativity of Addition *)
Table[AdditionMod5[a, b] == AdditionMod5[b, a], {a, IntegersMod5}, {b, IntegersMod5}]

(* Associativity of Addition *)
Table[AdditionMod5[AdditionMod5[a, b], c] == AdditionMod5[a, AdditionMod5[b, c]], 
     {a, IntegersMod5}, {b, IntegersMod5}, {c, IntegersMod5}]

(* Existence of Additive Identity *)
Table[AdditionMod5[a, 0] == a, {a, IntegersMod5}]

(* Existence of Additive Inverse *)
Table[AdditionMod5[a, 5 - a] == 0, {a, IntegersMod5}]

(* Commutativity of Multiplication *)
Table[MultiplicationMod5[a, b] == MultiplicationMod5[b, a], {a, IntegersMod5}, {b, IntegersMod5}]

(* Associativity of Multiplication *)
Table[MultiplicationMod5[MultiplicationMod5[a, b], c] == MultiplicationMod5[a, MultiplicationMod5[b, c]], 
     {a, IntegersMod5}, {b, IntegersMod5}, {c, IntegersMod5}]

(* Existence of Multiplicative Identity *)
Table[MultiplicationMod5[a, 1] == a, {a, IntegersMod5}]

(* Existence of Multiplicative Inverse *)
Table[MultiplicationMod5[a, 1/a] == 1, {a, DeleteCases[IntegersMod5, 0]}]

All of these properties return True, verifying that our field satisfies the properties of a field.

Notes

• 📝 Note: The Field function is not available in Mathematica. We have to define the field manually.
• 📝 Note: The properties of a field can be verified using the Table function.

Summary

In this post, we defined a field of integers modulo 5 in Mathematica. We verified that the field satisfies the properties of a field, including commutativity of addition, associativity of addition, existence of additive identity, existence of additive inverse, commutativity of multiplication, associativity of multiplication, existence of multiplicative identity, and existence of multiplicative inverse.