All Questions
9
questions
0
votes
1
answer
201
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Splitting field of $x^3 +x +1$ over $\mathbb F_{11}$
This is a HW problem for an algebra course.
Determine the splitting field of $f(x)=x^3+x+1$ over $\mathbb F_{11}$.
I tried to use the answers from this question and this question to help me, but want ...
- 2,454
1
vote
1
answer
50
views
If $F$ is a field $a\in F$, if $(x-a)^n$ for $n\geq 2$ divides $r(x)$ then $r(a)=r^{\prime}(a)=0$
Statment:
If $F$ is a field $a\in F$, if $(x-a)^n$ for $n\geq 2$ divides $r(x)$ then $r(a)=r^{\prime}(a)=0$.
Proof: Induction over $n$
Case $n=2$
Let $F$ be a field and $a\in F$ since $(x-a)^2 |r(x)$...
- 165
1
vote
2
answers
140
views
The irreducibility of $X^3-X^2+1$ over $\mathbb{F}_9[X]$
Let $K$ be the field with $9$ elements. I am asked to study the irreducibility of the polynomial $f:=X^3-X^2+1$ over $K[X]$.
I proceeded as follows. $\mathbb{Z}[i]/3\mathbb{Z}[i]$ is a field with $9$ ...
- 2,827
3
votes
1
answer
79
views
Solution Verification: Factoring $\left|\begin{smallmatrix}x&y&z\\x^p&y^p&z^p\\x^{p^2}&y^{p^2}&z^{p^2}\end{smallmatrix}\right|$ over $\mathbb{Z}_p.$
Problem: Factor $\begin{vmatrix} x & y & z \\ x^p & y^p & z^p \\
x^{p^2} & y^{p^2} & z^{p^2} \end{vmatrix}$ over $\mathbb{Z}_p$ as a
product of polynomials of the form $ax+by+...
- 5,230
0
votes
0
answers
52
views
Prove that each element of polynomial ring with irreducible characteristic has exactly one minimum polynomial.
I have the following problem:
Suppose $p$ is a prime and $f$ an irreducible polynomial in the ring $\Bbb{Z}/p\Bbb{Z}[X]$. Define a field $\mathbb{K}=\Bbb{Z}/p\Bbb{Z}[X]/(f)$ with the characteristic ...
- 425
0
votes
0
answers
40
views
multiplication of polynomials in $\mathbb{F}_2[x]$
Let $p(x) = 1 + x + x^2$ and $q(x) = 1 + x + x^3$. Then is the multiplication $p(x)q(x)$ obtained like this:
$$p(x)q(x)= (1 + x + x^2)(1 + x + x^3) = 1 +x +x^3 + x + x^2 + x^4 + x^2 + x^3 + x^5 $$
$$=...
- 1,862
1
vote
0
answers
285
views
Number of Roots of a Quadratic Modulo Prime Powers
I am an undergraduate and have been working on some research that I plan to publish. Part of the proof for one of my theorems relies on proving that an equation of the form $ax^2 + bx + c \equiv 0 \...
- 845
2
votes
1
answer
2k
views
Proving a polynomial is irreducible over a finite field
Question: $\mathbb{F}_5=\{0,1,2,3,4,5\}$ be the field with $5$ elements, let $\mathbb{F}_5[X]$ be the polynomial ring over $\mathbb{F}_5$. Let $m(X) = X^2+X+1$. Prove that $m(X)$ is irreducible over $\...
- 21
4
votes
1
answer
53
views
Find the degree of the polynomials in the following groups
Let $f(x) = x^4 + 6x^3 + 15x^2 + 10x + 1$ and $g(x) = 2x^2 + 15x + 1$.
Consider $f$ and $g$ as polynomials with coefficients in (a) $\mathbb Q$, (b) $\mathbb F_2$, (c) $\mathbb F_3$, and (d) $\mathbb ...
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