Math Problem Statement
Show that (π₯ + π¦) + π§ = π₯ + (π¦ + π§) for all π₯, π¦, π§ β π π
Solution
To prove that for all , we need to show that the operation of addition in the finite field is associative.
Step 1: Define the Elements
Let be elements of the finite field . In , addition is defined modulo . Therefore, for any two elements , the addition is given by the usual integer addition followed by taking the result modulo .
Step 2: Apply Addition Modulo
We want to prove the associative property of addition:
Let's consider both sides of this equation.
Left-Hand Side:
Now, add to the result:
Right-Hand Side:
Now, add to the result:
Step 3: Use Properties of Modular Arithmetic
In modular arithmetic, addition is associative. Therefore: [ [(x + y) \mod n + z] \mod n = [x + (y + z) \mod n] \mod n ] This simplifies to: [ [(x + y + z) \mod n] = [(x + y + z) \mod n] ]
Step 4: Conclusion
Since both sides are equivalent, we have shown that: for all .
Summary
The operation of addition in the finite field is associative, and thus holds true for any elements in .
Do you want more details or have any questions?
Related Questions:
- What is the definition of a finite field ?
- How does modular arithmetic work in ?
- What are the properties of addition in a finite field?
- How do you prove the commutative property of addition in ?
- Can you demonstrate the distributive property in ?
Tip: Associativity is a fundamental property of binary operations, and it's essential for the structure of fields and rings in abstract algebra.
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Math Problem Analysis
Mathematical Concepts
Associativity
Finite Fields
Modular Arithmetic
Formulas
-
Theorems
Associativity of Addition
Suitable Grade Level
Undergraduate Level
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