What Is the Minimal Polynomial of a=y^3 in the Field F=Kron(Z/2Z, x^4+x+1)?

[83956]

Homework Statement

Find the minimal polynomial of a=y^3 in F=Kron(Z/2Z, x^4+x+1). (Calculate the powers of a^2, a^3, and a^4.)

Homework Equations

The Attempt at a Solution

I attempted this trying to follow a similar worked problem in my book:

a=y^3 & y^4=y+1

Multiply by y^-3: y=y^-2 + y^-3

Plug in a

y=a+1

y^4+y+1 = 0 ... Multiply by y^-4: 1+y^-3+y^-4 = 0

Plug in a: a+1+a^-1 = a+1+a^2

So, a satisfies the irreducible polynomial x^2+x+1. Thus, each of the 16 elements of F can be written as a polynomial of degree at most 2 in a and a^2+a+1=0.
So, F=Kron(Z/2Z, a, x^2+x+1)

...did I do this correctly, or am I even close? I'm not sure of the relevance of calculating the powers of a^2, a^3, and a^4 as hinted in the problem statement.

 

[mighty2000]

Thank you for your question. It looks like you have made some progress in finding the minimal polynomial of a=y^3 in F=Kron(Z/2Z, x^4+x+1). However, there are a few errors in your solution.

Firstly, when you say "y^4=y+1", this is not correct. The correct equation should be "y^4=y^3+1". This is because in the field F=Kron(Z/2Z, x^4+x+1), x^4+x+1 is the minimal polynomial of y, not y^3.

Secondly, when you multiply by y^-3, you should get y^-2=y^-1+1, not y^-2=y^-3+1.

Thirdly, when you plug in a for y, you should get a^4=a^3+1, not a^4=a+1.

Now, to find the minimal polynomial of a=y^3, we can follow a similar approach as you did. We know that y^4=y^3+1 is an irreducible polynomial in F, and since a=y^3, we can plug that in to get a^4=a^3+1. This means that a satisfies the polynomial x^2+x+1, which is of degree 2.

To find the minimal polynomial of a, we need to find a polynomial of degree 2 that a satisfies. We can do this by calculating the powers of a^2, a^3, and a^4. We already know that a^4=a^3+1, so we can substitute a^3+1 for a^4 in this equation to get a^4=a^3+(a^3+1)=2a^3+1. Now, we can substitute a^3 for a^4 in this new equation to get a^4=2a^3+(a^3+1)=3a^3+1. Continuing this process, we can find that a^4=4a^3+1, a^4=5a^3+1, and so on. This means that a^4=na^3+1 for all positive integers n.

Now, we can use this information to find the minimal polynomial of a. Since a^4=na^3+1, we can rearrange this