1. ## finite field

I am going to ask a question this time. I have been trying to study some

abstract algebra, and I must admit it is some wacky stuff.

I respect NCA, PH, and others who excel at it.

Here is the question:

Show that {0,1,x,x^2} forms a field where 0 is the identity for addition and 1 is the multiplicative identity, and where 1+1=0, x^2=x+1.

Huh?. 1+1=0?. That is completely counter-intuitive. Should I try to construct a group table showing a+b=? and ab=?.

What is a good way to tackle something like this?. I want to get a handle on the basics before I try anything more advanced, that's for sure.

2. Originally Posted by galactus
I am going to ask a question this time. I have been trying to study some

abstract algebra, and I must admit it is some wacky stuff.

I respect NCA, PH, and others who excel at it.

Here is the question:

Show that {0,1,x,x^2} forms a field where 0 is the identity for addition and 1 is the multiplicative identity, and where 1+1=0, x^2=x+1.

Huh?. 1+1=0?. That is completely counter-intuitive. Should I try to construct a group table showing a+b=? and ab=?.

What is a good way to tackle something like this?. I want to get a handle on the basics before I try anything more advanced, that's for sure.
You could do the multiplication tables, it's the most intuitive way but it's tedious. I'd do it like this:

Let $A=\{ 0,1,x,x^2 \}$ and define $f:A \rightarrow \mathbb{Z}_2 \times \mathbb{Z}_2$ where $f(0)=(0,0)$, $f(1)=(1,1)$, $f(x)=(1,0)$, $f(x^2 )=(0,1)$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ is considered as an additive group (with operations coordinate-wise) and $g:A- \{0 \} \rightarrow \mathbb{Z}_3$(this last considered as an additive group) and $g(1)=0$, $g(x)=1$, $g(x^2 )=2$ It's clear that $f$ and $g$ define bijections, and there is a theorem that says: If $h:B \rightarrow G$ is a bijection where $B$ is a set and $G$ a group, both finite, then there is a unique function $*:B \times B \rightarrow B$ such that $(B,*)$ is a group isomorphic to $G$ and makes $h$ an isomorphism. Such operation is given by $a*b=f^{-1}(f(a)f(b))$, $a^{-1}=f^{-1}(f(a)^{-1})$ and $e_B=f^{-1}(e_G)$. It is straightforward to show that this operation induced by $f$ and $g$ on $A$ coincide with the sum and product you have already defined and as such are groups (abelian groups). Now you only have to check that the product distributes over the sum.

3. I feel embarrassed to ask this, but how would one build the table?. Even though, it is more tedious.

4. Originally Posted by galactus
I feel embarrassed to ask this, but how would one build the table?. Even though, it is more tedious.
The addition table looks like this:

$\begin{array}{c|cccc}+&0&1&x&x^2\\ \hline0&0&1&x&x^2\\ 1&1&0&x^2&x\\ x&x&x^2&0&1\\x^2&x^2&x&1&0\end{array}$

(Notice that $1+x^2=1+(1+x) = 0+x = x$. Also $x+x = x(1+1) = x\cdot0 = 0$; and $x+x^2 = x + (x+1) = 0+1 = 1$.)

Do the multiplication table similarly, noticing that $x\cdot x^2 = x(x+1) = x^2+x = 1$ and $x^2\cdot x^2 = (1+x)^2 = 1+x^2 = x$.

5. Wow, thanks a lot, Opalq. I have always wanted to learn some algebra, but was afraid to put in the effort.

6. One way of proving that it's a field is just noticing that it's $\mathbb{F}_2[X]/(X^2+X+1)$. Since $X^2+X+1$ is an irreducible polynomial, the quotient ring $\mathbb{F}_2[X]/(X^2+X+1)$ is a field.

7. Coincidentally, I was actually looking that irreducible polynomial up, Bruno.

8. if you have a little bit background in field theory, then the easiest way would be to use this result that "the" finite field of order $q=p^n,$ where $p$ is a prime, is exactly the set of the roots of the

polynomial $f(t)=t^q - t \in \mathbb{F}_p[t].$ then you'd only need to prove that the elements of your set are the roots of the polynomial $f(t)=t^4 - t,$ viewed as an element of $\mathbb{F}_2[t],$ which is very easy:

obviously $f(0)=f(1)=0.$ also, since $x^2=x+1,$ we have $x^4=x$ and thus $f(x)=x^4-x=0.$ finally from $x^4=x$ we get $x^8=x^2$ and thus $f(x^2)=x^8 - x^2=0.$

9. Thanks NCA, but I have no background in this stuff. That is why I am trying to learn a little. I figured you all would be the best place to start.

I don't even know what $F_{p}[t]$ stands for.

I do not expect you to give me an entire lesson. Thanks for your input.

I have a book...Abstract Algebra, 2nd edition by Hungerford.