Tagged Questions
1
vote
0answers
30 views
$X^n-a \in k[X]$ ,char(k)|n. Multiplicity of roots of irreducible polynomial which devides $X^n-a$.
Let $k$ be a field of $char(k)=p>0$, $f(X)=X^n-a \in k[X](a \neq 0)$, $p|n$.
If $g(X)$ is an irreducible polynomial in $k[X]$ and $g(X)|f(X)$, do all the roots of $g(X)$ have the same ...
6
votes
3answers
74 views
Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies
Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime.
Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
1
vote
0answers
37 views
Showing $f(x)=g(x^{p^a})$ over field of (prime) characteristic $p>0$.
Let $f$ be a non-constant irreducible polynomial over a field $F$ of (prime) characteristic $p>0$. I need to prove that $f$ can be presented as:
$f(x)=g(x^{p^a})$, where $g$ is irreducible over ...
1
vote
2answers
86 views
Roots of polynomials over finite fields
I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$.
I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
0
votes
1answer
39 views
Polynomials decomposition into irreduceables
I've been trying to find the composition to irreduceables of the following polynomials with no much success:
X^2 +1 over the field F7
and X^2-2 over the field F5
Is there any method/algorithms I ...
1
vote
2answers
37 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
1
vote
2answers
56 views
Is $x^2+1$ irreducible over a cyclotomic field?
Let $K=\mathbb{Q}[\omega]$, where $1+\omega+\omega^2=0$, let $f(X)=X^2+1$. How can i prove irreducibility of $f$ over $K$?
5
votes
6answers
134 views
Show that $x^4 +1$ is reducible over $\mathbb{Z}_{11}[x]$ and splits over $\mathbb{Z}_{17}[x]$.
Reduction into linear factors $\mathbb{Z}_{17}[x]$:
This part is not too hard: $x^4 \equiv -1$ mod 17 has solutions: 2, 8, 9, 15 so
$(x-2)(x-8)(x-9)(x-15) = x^4 -34 x^3 +391 x^2-1734 x+2160 \equiv ...
0
votes
2answers
83 views
Number of irreducible polynomials of degree $3$ over $\mathbb{F}_3$ and $\mathbb{F}_5$.
I'm trying to find the number of $3$rd degree irreducible polynomials over $\mathbb{F}_3$ and $\mathbb{F}_5$.
Since a $3$rd degree polynomial is irreducible if and only if it is divisible by a ...
1
vote
2answers
44 views
Convert polynomials and fractions in a finit field?
I am trying to understand how finite field works, and I am stuck on converting high power polynomials into a power of the field, also converting fractions into integers.
$8^{-1}\cdot44$ in $\Bbb ...
2
votes
1answer
78 views
Roots of a $p$-polynomial all have same multiplicity $p^e$.
This is a problem for fun from Basic Algebra I, by Nathan Jacobson. (3 pp. 234.)
Let $F$ be of characteristic $p$. A polynomial $f(x)\in F[x]$ is called a $p$-polynomial if it has the form ...
2
votes
3answers
79 views
Let $F$ be a field, then the polynomial $x^n - 1$ has $n$ roots in $F$ if $F$ contains a multiplicative subgroup of order $n$.
Is this following true? If $F$ is a field, then the polynomial $x^n - 1$ has $n$ roots in $F$ whenever $F$ contains a multiplicative subgroup of order $n$.
4
votes
1answer
50 views
Find separable irreducible $g$ such that $f(x)=g(x^{p^d})$
This is an exercise from VII.4. in Algebra: Chapter 0.
Let $\mathcal{k}$ be a field of characteristic $p$, and $f(x)\in\mathcal{k}[x]$ an inseparable irreducible polynomial. Find a separable ...
0
votes
1answer
61 views
On a characterization of primitive polynomials over a finite field
Let $K$ be a finite field. Let us define a primitive polynomial as an $f \in K[X]$ s.t. the multiplicative order of $X$ in $K[X]/(f)$ is equal to $|K|^{\deg f} - 1$. I want to show that $f \in K[X]$ ...
4
votes
1answer
49 views
Polynomial Fields.
I am trying to find a polynomial in $\Bbb{Q}[x]$ which is irreducible over $\Bbb{Q}$ and has at least one linear factor over $\Bbb{R}$ and at least one irreducible quadratic factor over $\Bbb{R}$. Any ...
4
votes
4answers
111 views
Why do $f$ and $f'$ generate all of $K[X]$?
I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258.
He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
4
votes
1answer
97 views
Multiple roots of polynomials over a finite field
Show that $x^4+x+1$ over $\mathbb{Z}_2$ does not have any multiple zeros in any field extension of $\mathbb{Z}_2$.
Show that $x^{21} + 2x^8 +1$ does not have multiple zeros in any ...
3
votes
4answers
78 views
Prove that $f\sim g$ iff $f\mid g$ and $g\mid f$, where $f$ and $g$ are polynomials.
Prove that $f\sim g$ iff $f\mid g$ and $g\mid f$, where $f$ and $g$ are polynomials.
It seems to me that $f$ and $g$ must be equal.
And if you guys could direct me to sources that explain ...
3
votes
2answers
139 views
Can nonzero polynomials vanish identically?
I know that a nonzero single-variable polynomial over a finite field can vanish identically e.g. take the product $\prod_a(x-a)$ for every $a$ in the field. But I know that for an infinite field this ...
5
votes
4answers
193 views
Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.
Show $p(x) = x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.
By Gauss' Lemma, $p(x)$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in $\mathbb Z[x]$. We can look at ...
0
votes
2answers
46 views
Two different definitions of separable polynomial
This is from A field guide to Algebra by Antoine Chambert Loir.
A polynomial $P \in K[X]$ is separable if and only if its roots are in an algebraic closure of $K$ are simple.
Here is ...
3
votes
2answers
101 views
Proving $\sqrt[6]{2}\not\in\mathbb{Q}(\sqrt[3]{2})$
Can someine please tell me quickly how to prove that $\sqrt[6]{2}\not\in\mathbb{Q}(\sqrt[3]{2})$ ? I have an exam in a about an hour and I can't figure this out (my idea was to prove it indirectly, ...
0
votes
1answer
43 views
Extensions of $\sigma:\mathbb{Q}(\xi)\rightarrow\mathbb{Q}(\xi), \ \sigma(\xi)=\xi^{2}$ to $\mathbb{Q}(\xi)(\sqrt[3]{2})$
Let $L$ be the splitting field of $T^{3}-2$ and let $\sigma:\mathbb{Q}\left(\xi\right)\rightarrow\mathbb{Q}\left(\xi\right)$
be a morphism defined by $\sigma\left(\xi\right)=\xi^{2}$. Find all
...
3
votes
2answers
59 views
Find intermediary extension
Let $P$ be the polynomial
$$
P=64X^9 + 192X^7 + 240X^6 + 36X^5 + 552X^4 - 847X^3 - 540X^2 + 273X + 143
$$
Then $P$ is irreducible over $\mathbb Q$, and has three real roots. Let $\alpha$ be one of ...
5
votes
2answers
138 views
Closure of a number field with respect to roots of a cubic
Consider the following property of subfields ${\mathbb K}$ of ${\mathbb C}$ :
$$
\text{Any polynomial of degree 3 with coefficients in} \ {\mathbb K} \ \text{has a root in } {\mathbb K} \ \ \ \ ...
2
votes
0answers
65 views
The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$
I'm trying to solve this question:
Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a
field $F$.
Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the
...
1
vote
0answers
121 views
Irreducibility of $x^{p^r} - a$ if $a$ is not a $p$-th power.
Let $F$ be a field of characteristic 0, $p$ an odd prime and $a\in F$ not a p-th power. Then $x^p-a$ is irreducible over $F$. To prove this we assume the contrary, put $\alpha\not\in F$ a root of some ...
0
votes
4answers
368 views
How can I know how many real roots this polynomial has?
Let $x^7-10x^5+15x+5$ a polynomial over $\mathbb Q$. I would like to know how many real roots this polynomial has, I know we have to use the intermediate value theorem, but I don't know how to use in ...
1
vote
1answer
80 views
Steps missing in the proof $[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible}$
Consider the following
THEOREM Let $f\in K[X]$ have degree $n$ and splitting field $L/K$. Then we
have $$[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible}$$
and its
Proof $\ $ Suppose ...
6
votes
2answers
195 views
$[L:K]=n!\ \Longrightarrow \ f$ is irreducible and $\text{Gal}(L/K)\cong S_n.$
How do I go about proving the following theorem ?
Let $f\in K[T]$ have degree $n$ and splitting field $L/K$. Then we have $$ [L:K]=n!\ \Longrightarrow \ f\text{ is irreducible and Gal}(L/K)\cong ...
0
votes
2answers
122 views
Doubts in the fundamental theorem of algebra using Galois theory
I'm studying the Bhattacharya's algebra book, and I have the following doubts:
1-What this $g(x)$ has to do with $f(x)$? I mean why proving the splitting field of $g(x)$ over $\mathbb R$ is $\mathbb ...
3
votes
4answers
161 views
First examples in Galois theory
I'm studying Field Theory and after studying theorems and problems about extensions, splitting fields, etc... I'm starting with the first theorems of the Galois Theory itself. In order to see if I ...
5
votes
2answers
211 views
Irreducibility of polynomial if no root (Capelli)
Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$.
This answer to a related question ...
4
votes
3answers
113 views
The irreduciblity of $X^{2r}-X^{r}+1$
The question is that $e^{2\pi i/6r}$ is a root of the polynomial $X^{2r}-X^{r}+1 \in \mathbb Q[X]$ , we want to prove that $X^{2r}-X^{r}+1$ is irreducible if and only if $r$ is of the form ...
16
votes
1answer
278 views
Irreducibility of $x^{n}+x+1$
Motivated by this problem, and KCd's comment on my answer, I am left with the following question:
Question: Suppose that $n\not \equiv 2\pmod{3}$. Is $$x^n+x+1$$ irreducible over $\mathbb{Q}$?
...
2
votes
1answer
96 views
Questions about a vector space over a finite field with a bilinear symmetric form.
This is an extension of a previously asked question: Inner Product Spaces over Finite Fields.
Inner product spaces in the typical undergraduate linear algebra course are stressed to be over ...
2
votes
0answers
37 views
Does this kind of field extension degree have a name?
Suppose we have a finite, normal field extension $K\subseteq L$. Then $L$ is the splitting field of a polynomial over $K$.
Now, consider $n$, the minimal degree of a polynomial for which $L$ is the ...
4
votes
4answers
87 views
Show that over any field $K$, such that $\mathbb Q \subset K$ the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors
Show that over any field $K$, such that $\mathbb Q \subset K$ the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors
I think if we prove that: if $K$ contain one root it ...
2
votes
2answers
79 views
Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b $ has degree of extension 3
Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b \in \mathbb Q[x]$ has degree of extension 3 over $\mathbb Q$.
I'm trying solve do this question, it seems very difficult to ...
1
vote
1answer
73 views
Polynomial of degree $n$ over a field of characteristic $p>0$ has at most $n/p$ distinct roots
Let $f$ be a polynomial of degree $n$ over a field $F$ of characteristic $p$. Suppose $f'=0$. Show that $p\mid n$ and that $f$ has at most $n/p$ distinct roots.
I can't solve this question, any help ...
1
vote
2answers
86 views
Find the smallest normal extension
Find the smallest normal extension (up to isomorphism) of $\mathbb Q(\sqrt[4]2)$ in $\overline {\mathbb Q}$ (the algebraic closure of $\mathbb Q)$
My atempt:
$\mathbb Q(\sqrt[4]{2},i)$ is a normal ...
3
votes
3answers
208 views
The splitting field of $x^3+x^2+1$ over ${\Bbb Z}/(2)$
Let $F=\mathbb Z/(2)$. The splitting field of $x^3+x^2+1\in F[x]$ is a finite field with eight elements.
my attempt of solution:
If $\alpha$ is a root in this polynomial in its splitting field, then ...
5
votes
2answers
245 views
How do I prove that this polynomial is irreducible?
How do I prove that $x^4+1$ is an irreducible polynomial over $\mathbb Q$? I've already tried the Eisenstein criterion which gives to me any results to solve this question, I need help here.
Thanks
4
votes
1answer
145 views
Showing a polynomial not reducible.
How do I show that $f(x)=1+2x+\cdots+(p-1){x}^{p-2}$ is not reducible on $\mathbb{Q}$, where $p$ is prime.
0
votes
3answers
42 views
$x^3-\alpha \in \Bbb Q(\alpha)[x]$ is irreducible
Given $\alpha\in \Bbb C$ trascendental , and such that $|\alpha|=1$ (I don't know if this is necesary but I need only this case). Then I have to prove that the polynomial $x^3-\alpha \in \Bbb ...
3
votes
1answer
156 views
the discriminant of the cyclotomic $\Phi_p(x)$
I'm very bad in computations of this kind :/. I don't know tricks for computing the discriminant of a polynomial, only the definition and using the resultant, but it's very complicated to do only with ...
0
votes
1answer
58 views
counting the real zeros of a polynomial and proving that it's irreducible over $\Bbb Q$
Let's consider the polynomial $$
f\left( x \right) = \left( {x^2 + 2} \right)\prod\limits_{i = - k}^k {\left( {x - 2i} \right) + 2 \in {\Bbb Q}\left[ x \right]}
$$ . Let's suppose that $
p = 2k + ...
7
votes
1answer
95 views
“Real part” of a number field
Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
2
votes
1answer
56 views
Construction of a field
Given the polynomial
$$f(x)= x^4-16x^2+4$$
which has $a=\sqrt 3+\sqrt 5$ as one of its roots in $\Bbb C$, can you use $f(x)$ to construct a field $E$ of the form $Q[x]/I $ for some appropriate ideal ...
0
votes
2answers
65 views
Relation between torsion subgroup of multiplicative group of field and solvability of polynomials
In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
