All Questions
12
questions
0
votes
2
answers
81
views
Is this $f(x) = x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ irreducible in GF(5)?
Perhaps one can somehow apply Eisenstein's sign here by considering $f(x+1)$, but by default it is formulated for the expansion over $\mathbb{Q}$ of a polynomial from $\mathbb{Z}[x]$. Here we have $GF(...
- 15
5
votes
1
answer
261
views
Galois group of the polynomial $x^n+x^{n-1} +⋯+x^2+x−1$
As we already know, the following polynomial is irreducible over $\mathbb Q[x]$:
$$x^n + x^{n-1} + \cdots + x^2 + x - 1 = \frac{x^{n+1} - 2x + 1 }{ x-1}.$$
By Descartes' rule of signs, it has only 1 ...
- 630
0
votes
0
answers
64
views
What does the 3rd volume of Lehrbuch der Algebra of Heinrich Weber talk about? [duplicate]
I don't know the German language and I stumbled upon this book. I tried google translate and utterly failed. I just need to know what this volume is mainly about.
- 1
11
votes
1
answer
531
views
Can two monic irreducible polynomials over $\mathbb{Z}$, of coprime degrees, have the same splitting field?
Let $f,g \in \mathbb{Z}[X]$ be monic polynomials. It is possible for distinct monic polynomials over $\mathbb{Z}$ to have the same splitting field. For example $f = x^4 - 2$ and $g= x^4+2$ both have ...
- 113
0
votes
0
answers
54
views
Notation and interpretation of Polynomials in $\mathbb{F}_{p}[x]$
i'm confused with some notation that involves reduction of polynomyals on $\mathbb{Z}[x]$ to $\mathbb{F}_p[x]$. It's part of the proof that Cyclotomic polynomials are irreducible over $\mathbb{Q}[x]$.
...
- 1,403
1
vote
1
answer
59
views
Unique subfield of order $2$ of $\mathbb{Q(\zeta_{7}})$ over $\mathbb{Q}$
Actualy this subfield is $\mathbb{Q}(i\sqrt{7})$ since $X^{2}+7$ is irreducible over $\mathbb{Q}$ and has $i\sqrt{7}$ as root.
My problem here is to show unicity, I tried something using the tower ...
- 1,403
2
votes
2
answers
44
views
Working with polynomials in $\mathbb{Z}_{2}[X]$, finding roots and splitting fields
I'm a beginner in the area and yet can't see how to work with a polynomial in other fields diferents from $\mathbb{Q}$.
I have the polynomial $f(X)=X^5 -X^2+1 \in \mathbb{Z}_{2}[X]$ and must prove ...
- 1,403
2
votes
1
answer
681
views
For $p$ prime, is the polynomial $x^p-x+1$ irreducible in $\mathbb{Z}_p$? [duplicate]
It is possible, for $p\in\mathbb{N}$ prime, that the polynomial
$x^p-x+1$ is irreducible in $\mathbb{Z}_p$?
By the identity
$a^p\equiv a$ mod $p$ for any $a\in \mathbb{Z}_p$ surely there is not a ...
- 8,856
6
votes
1
answer
367
views
Splitting field of $f(X)=X^4-X^3-5X+5$ and its degree.
Let $K$ be a splitting field in $\mathbb C$ of the polynomial $f(X)=X^4-X^3-5X+5$ over $\mathbb Q$.
Construct the splitting field $K$ and find the degree of the extension $K:\mathbb Q$.
$f(...
user484410
1
vote
1
answer
117
views
Decomposing $x^4-5x^2+6$ over some fields
My book asks me to decompose
$$x^4-5x^2+6$$
over:
$K = \mathbb{Q},\\ K = \mathbb{Q[\sqrt{2}]},\\ K = \mathbb{R}$
For $K = \mathbb{Q}$, I substituted $x² = a$ to get:
$$a²-5a+6 = (a-3)(a-2)$$
So ...
- 6,756
0
votes
1
answer
307
views
Galois Group of a cubic polynomial
Let $P=X^3+a_1X^2+a_2X+a_3 \in \mathbb{Q}[X]$ be irreducible, $x_1,x_2,x_3$ the roots of $P$ and $L:=\mathbb{Q}[x_1,x_2,x_3]$. The galois group of $P$ is isomorph to $S_3$.
Now we define $z:=(x_1-x_2)...
- 101
1
vote
1
answer
232
views
Is this a generator of a cyclic group?
Let $F=\mathbb Z/(p)$, where $p$ a prime number, $f(x)$ a monic irreducible polynomial in $K=F[x]$ of degree $n$, $K=F[x]/(f(x))$, and $E$ the multiplicative group of nonzero elements of $K$. Then ...
- 157