Prove Polynomials of Degree 1, 2 & 4 Have Roots in Z_2[x]/(x^4+x+1)

In summary, The conversation discusses finding a root in Z_2[x]/(x^4+x+1) for every polynomial of degree 1, 2, and 4 in Z_2[x]. The use of latex commands is explained in the introduction to LaTeX thread. The speaker suggests trying all possible cases for irreducible polynomials and finding a general condition for why polynomials of degree 1, 2, and 4 have roots, while one of degree 3 does not. They also emphasize the importance of approaching mathematics by examining specific cases and finding patterns.
  • #1
whodoo
4
0
I want to show that every polynomial of degree 1, 2 and 4 in Z_2[x] has a root in Z_2[x]/(x^4+x+1). Any ideas?

Ps. How can I use latex commands in my posts?
 
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  • #2
You can just do it by exhaustion. There are very few cases to think about (and if you have to start thinking about the case deg=1 then you're missing something).

LaTeX is explained in the introduction to LaTeX thread (probably in physics): just look around.
 
  • #3
Okey, something like this?
The case deg 1 is obvious. The case 2 is obvious except for the irreducible x^2+x+1, where I can find a solution. Degree 4 is obvious except for the irreducible cases which I can try?

Im not really satsified with this solution.
 
  • #4
Why not? Have you actually tried to look at the irreducible casees? If you do you might figure out what a general condition would be, and why the degs 1,2,4 polys all satisfyit (and presumably why there is a deg 3 poly that doesn't). You won't figure out a general criterion just by pulling it out of thin air. You look at cases where it is true, where it isn't and figure out what quantifies the difference. That's how you do maths.
 

What is the definition of a polynomial of degree 1, 2, or 4?

A polynomial of degree 1, 2, or 4 is a mathematical expression consisting of one or more terms with non-negative integer exponents and coefficients. The degree of a polynomial is determined by the highest exponent present in the expression.

What is Z_2[x]/(x^4+x+1)?

Z_2[x]/(x^4+x+1) is a quotient ring, also known as a residue class ring, formed by taking the set of polynomials with coefficients in the field Z_2 (the integers modulo 2) and dividing by the polynomial x^4+x+1.

What does it mean for a polynomial to have a root?

A root of a polynomial is a value that, when substituted into the polynomial, results in an output of 0. In other words, a root is a solution to the polynomial equation.

Why is it important to prove that polynomials of degree 1, 2, and 4 have roots in Z_2[x]/(x^4+x+1)?

Proving that polynomials of degree 1, 2, and 4 have roots in Z_2[x]/(x^4+x+1) is important in the study of algebraic structures and their properties. It also has applications in coding theory and cryptography.

What is the significance of the field Z_2 in this context?

The field Z_2, also known as the binary field, consists of only two elements (0 and 1). This makes it a useful tool for studying and analyzing mathematical structures, as it simplifies calculations and allows for easier proofs. In the context of proving that polynomials of degree 1, 2, and 4 have roots in Z_2[x]/(x^4+x+1), Z_2 is used as the underlying field for the quotient ring.

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