(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.)

2. Relevant equations

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Finding a Minimal Polynomial

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**