# Tag Info

Accepted

### How to factor a polynomial quickly in $\mathbb{F}_5[x]$

The observation made by user8268 is the key to the following paper and pencil factorization. Not ruling out the existence of other possibilities — that trick simply works like charm. Let $\alpha$ be ...
• 132k
Accepted

### number of rational points of hyper elliptic curve $y^5=-x^2+x$ over $\Bbb{F}_{121}$

The job is easily done with computer support, here sage. Let us do this first. ...
• 32.3k

### Distribution of Primitive Elements Finite Fields Prime Order

Gauss proved that the sum $S_p$ of all primitive roots modulo $p$ in the interval $[1,p-1]$ is congruent to $\mu(p-1)$ modulo $p$. This result appeared several times before in MSE and a possible proof ...
• 19.9k
Accepted

### Roots of $x^4+x^3+x^2+x+1$ over $\mathbb{Z}_3[x]/(x^3-x+1)$.

Using the same notation, $x$, for both as the variable of the polynomial and the field leads to the mistake. The question is the following: Let $\mathbb{F}=\mathbb{Z}_3[x]/(x^3-x+1)$. Show that the ...
• 4,812
Accepted

### Representing finite fields

Zech Logarithms are useful for performing addition when you represent elements as powers of a multiplicative generator. They depend on a choice of the minimal polynomial of the generator. Although all ...
• 89.3k

### Irreducible polynomials in the field with 4 elements

You are searching for irreducible polynomials of degree $\le3$, so the elementary irreducibility test works: a quadratic or a cubic $f(x)\in\Bbb{F}_4[x]$ is irreducible if and only if it has no zeros ...
• 132k

### Need example of a finite noncommutative ring with inverses

A ring is called a division ring if any non-zero element is invertible. A field is a commutative division ring. So what you are asking for is a finite division ring which is not a field. Actually such ...
• 1,218
Accepted

### $𝜑(𝑥) = 𝑥^{𝑛−1} + 𝑥^{𝑛−2} + ⋯ 𝑥 + 1$ then $(𝜑(𝑥))^2 = 𝑛𝜑(𝑥)$ in $𝔽[𝑥]/⟨ 𝑥^𝑛 − 1 ⟩$

This problem rewrites $$\left(\frac{x^n-1}{x-1}\right)^2\equiv n\frac{x^n-1}{x-1}\bmod{x^n-1}$$ i.e. $$x^n-1\equiv n(x-1)\bmod{(x-1)^2}$$ or equivalently $$(1+y)^n-1\equiv ny\bmod{y^2}.$$ This follows ...
• 33.4k
Accepted

### Is $x^{100} - x^2 + 1$ separable in an algebraic closure of $\mathbb{F}_2$

The fact that your polynomial has no root does NOT imply that it is irreducible (this kind of argument works only for polynomials of degree $2$ or $3$. Think of $(x^2+1)^2\in\mathbb{R}[x])$. (And your ...
• 14.8k
Accepted

### "Low degree Polynomials do not have too many roots" - what exactly does this mean?

It seems to me like "low degree polynomials don't have too many roots" is a kind of slogan which is meant to be memorable. This slogan is made precise by the statement "A (nonzero) ...
• 37.9k
Accepted

• 2,665