Articles of finite fields

Finding a primitive element of a finite field

Let $F = \mathbb{F}_p[x]/(m(x))$, where $m(x)$ is irreducible in $\mathbb{F}_p[x]$. How do I find a primitive element of $F$, i.e., one that generates the nonzero elements of $F$ multiplicatively? For example, in $\mathbb{F}_{256} = \mathbb{F}_2[x]/(x^8+x^4+x^3+x+1)$, the element $x+1$ has order $255$, so it is a primitive element of $\mathbb{F}_{256}$.

How to prove this is a field?

Let $F=(\mathbb{Z}/5\mathbb{Z})[x]/(x^2+2x+3)$. How do I prove $F$ is a field? I’ve shown its a commutative ring with an identity $\bar1$. Then we let $(\bar{a}x+\bar{b})^{-1}=(\bar{c}x+\bar{d}).$ Multiplying those together gives me and substituting $3x+2$ for $x^2$ gives me the following two equations…not sure what to do next. $\bar{3ac}+\bar{ad}+\bar{ac}=\bar0$ $\bar{2ac}+\bar{ad}=\bar1$ I also need to prove that every element […]

Show $F(U) = K((x^q -x)^{q-1})$.

Let $K$ be a finite field with $q$ elements. Show that if $U$ is the subgroup of $Aut(K(x)/K)$ which consists of all mappings $\sigma$ of the form $(\sigma \theta)(x) = \theta(ax+b)$ with $a \neq 0$ then $F(U) = K((x^q -x)^{q-1})$. I am not getting any clue to solve the problem. Help Needed. Here $F(U)$ is […]

Field extensions and monomorphisms

I am working through some algebra exercises and got stuck with the following problem: I am working in the finite field $\mathbb{Z}_5$. Let $f \in \mathbb{Z}_5[x]$ and $f(x) = x^2 + 2$. By simple trial and error, I checked that this polynomial is irreducible, hence $\mathbb{Z}_5/(f) \cong GF(5^2)$, denoting the Galois field ($f$ is of […]

Wiedemann for solving sparse linear equation

I am new member. I am researching in Wiedemann algorithm to find solution $x$ of $$Ax=b$$ Firstly, I will show a Wiedemann’s deterministic algorithm (Algorithm 2 in paper Compute $A^ib$ for $i=0..2n-1$; n is szie of matrix A Set k=0 and $g_0(z)=1$ Set $u_{k+1}$ to be $k+1$st unit vector Extract from the result of step […]

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?

Matrix multiplication using Galois field

$$\begin{bmatrix}1 &1 &6\\4& 3& 2\\5 &2& 2\\5& 3& 4\\4& 2& 4\end{bmatrix}\begin{bmatrix}4\\5\\6\end{bmatrix} = \begin{bmatrix}3\\5\\4\\3\\2\end{bmatrix}. $$ I am not getting that how come this result is possible ? [Editor’s comment #1: The question makes sense, but the asker forgot to explain their notation – possibly because they have not been exposed to any alternatives (happens regrettably often […]

Some iterate of a linear operator over $\mathbf F_q$ is a projection

If $T$ is an endomorphism of a finite-dimensional vector space $V$ over a finite field, then how can I show that there exists a positive integer $r$ such that $T^r$ is a projection operator?

Finite field satisfies $1+\lambda^{2}-\alpha\mu^{2}=0$

Lemma. If $F$ is a finite field, $\alpha\neq0\in F$ then there exists $\lambda,\,\mu\in F$ so that $1+\lambda^{2}-\alpha\mu^{2}=0$. Proof. If the characteristic of $F$ is $2$, $F$ has $2^n$ elements and every element $x$ in $F$ satisfies $x^{2^n}=x$. This every element in $F$ is a square. In particular $\alpha^{-1}=\mu^{2}$. Using this $\mu$ and $\lambda=0$ we get […]

For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2$?

For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$ ? E.g. $x+1$, $x^2+x+1$ are irreducibles. Subcase of this question Factor by irreducible is field. Does it help ?