Example of algebras over GF(2)

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In summary, the conversation discussed finding an example of an algebra over the field \mathbb{Z}_2 with specific properties. One suggestion was \mathbb{Z}_2[X]/(X^3), but it was pointed out that this algebra does not have unity. Other examples were suggested, including taking all polynomials over two variables modulo the relation x^3=y^3=0 or (x)(y^2)=(x^2)(y).
  • #1
Lie
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Anyone know of an example of an algebra over the field [tex]\mathbb{Z}_2[/tex] with the following properties?
1. commutative;
2. associative;
3. [tex] x^3 = 0 [/tex], for all x; and
4. Exists x and y such that [tex] x^2y \neq 0 [/tex].

Grateful!
 
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  • #2
What about [tex]\mathbb{Z}_2[X]/(X^3)[/tex]? It satisfies your first three properties, and alse the last one with y=1 and x=X...
 
  • #3
micromass,

Note that condition 3 implies that the algebra can not have unity. Therefore [tex] \mathbb{Z}_2[X]/(X^3) [/tex] is not an example.
 
  • #4
Oh sorry, I forgot to read "for all x" :frown: Well, I'll look for another example...
 
  • #5
What goes wrong with the direct way to approach the problem? (e.g. like micromass's, except working with algebras rather than rings)
 
  • #6
Ermm, can't you just take all polynomials over two variables x and y modulo the relation x^3=y^3=0 ?
 
  • #7
Jamma said:
Ermm, can't you just take all polynomials over two variables x and y modulo the relation x^3=y^3=0 ?

Jamma, same remark:
Lie said:
micromass,

Note that condition 3 implies that the algebra can not have unity.[...]
 
  • #8
Sorry, I didn't mean it like that, I should describe my algebra a bit better.

Take as our set of elements {0,x,x^2,x^3,y,y^2,y^3} and all multiples and linear combinations of them with the obvious rules of addition and multiplication subject to the condition that x^3=y^3=0.

There is no unity here.

[Edit:ignore me, this algebra has elements in it which don't cube to zero]
 
Last edited:
  • #9
Ok, how about my algebra up there but with the relation (x)(y^2)=(x^2)(y).

It seems at a first glance that this works.
 

What is an algebra over GF(2)?

An algebra over GF(2), or Galois field of order 2, is a mathematical structure consisting of a set of elements, operations, and axioms defined over the binary field with 2 elements, 0 and 1.

What are some examples of algebras over GF(2)?

Some examples of algebras over GF(2) include Boolean algebras, vector spaces over GF(2), and finite fields of characteristic 2.

How are algebras over GF(2) used in cryptography?

Algebras over GF(2) are used in cryptography to encode and decode messages, as well as to perform operations such as encryption and decryption. They are also used in error-correcting codes and in the design of secure communication protocols.

What is the difference between an algebra over GF(2) and an algebra over a different field?

The main difference between an algebra over GF(2) and an algebra over a different field is the number of elements in the field. GF(2) only has 2 elements, while other fields can have infinitely many elements. This difference affects the structure and properties of the algebra.

What are the applications of algebras over GF(2) in computer science?

Algebras over GF(2) have many applications in computer science, including coding theory, error correction, and cryptography. They are also used in computer graphics and image processing algorithms, as well as in the design of error-free digital circuits.

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