MHB Elements of $\mathbb{F}_{2^2}: What Are They?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Elements
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hello! :o

Which are the elements of $\mathbb{F}_{2^2}$ ?? (Wondering)

There are $4$ elements, right??

For each element $a$ in $\mathbb{F}_{2^2}$ it stands that $a^2=a \Rightarrow a^2 \in \mathbb{F}_{2^2}$, right??
 
Physics news on Phys.org
The elements are the following:

$$0,1,a,b$$

The product of two elements is in the field, so $$a \cdot b=0 \text{ or } 1 \text{ or } a \text{ or } b$$

It cannot be $0$ because $\mathbb{F}_{2^2}$ is an integral domain and since $a , b \neq 0$ the product cannot be $0$.

The product cannot ba $a$ or $b$, because then one of $a$ and $b$ must be $1$, and then the field would contain only $3$ elements.

Therefore, the product must be $1$, that means that $a \cdot b=1 \Rightarrow
b=a^{-1}$.

So, the elements are $$0,1,a,a^{-1}$$ right?? (Wondering)

Also, how could we show that $$\mathbb{F}_{2^2} \cong \mathbb{Z}_2(a)$$ where $a$ is of degree $2$ over $\mathbb{Z}_2$?? (Wondering)

Do we have to show that these two fields have the same elements?? (Wondering)
 
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...
Back
Top