1
vote
3answers
43 views

An irreducible polynomial cannot share a root with a polynomial without dividing it

There is a lemma of Galois stating, "An irreducible equation can have no common root with a rational equation without dividing it". His definitions are a little bit imprecise, but I think he means: ...
5
votes
1answer
77 views

Difficult algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
0
votes
1answer
48 views

Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form ...
2
votes
2answers
47 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
6
votes
1answer
93 views

finite field extension problem

Maybe somebody knows how to proove the following algebraic theorem: $C \subset U$ is a field extension and $N \subset U$ so, that all $x \in N$ are algebraic over $C$ and $C[N]=\left\lbrace ...
4
votes
1answer
60 views

Field Theory, Factor Ring, Polynomials

I have the following problems: (1) Let $g=X^2+\overline{4}$ and $h=X^2+\overline{2}$ be polynomials in $(\mathbb{Z}/\mathbb{Z}7)[X]$. $L$ and $K$ are the splitting fields of $g$ and $h$ over ...
1
vote
1answer
32 views

If $[E:F]$ is finite and $\alpha \in E$ then there is an irr. polynomial in $F[x]$ with root $\alpha$

I'm studying for an exam and encountered a confusing proof of the following fact in my notes: Let $[E:F]$ be finite and $\alpha \in E$ then there is an irreducible polynomial $p(x) \in F[x]$ with ...
0
votes
3answers
27 views

For an irreducible polynomial $p$ and root $\alpha$, $[F[\alpha]:F]$ = degree of $p$

I'm studying for an exam, and I couldn't find the proof for the following theorem in my notes: For $p(x) \in F[x]$ irreducible and $\alpha$ a root of $p$ in some extension field, $[F[\alpha]:F] =$ ...
1
vote
0answers
24 views

The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
0
votes
1answer
21 views

irreducibility of $x^2-a$ in $\mathbb{Z}_2[a]$

Let $a$ be transcendental over $\mathbb{Z}_2$ and let $F=\mathbb{Z}_2(a)$. Prove $p(x)=x^2-a$ is irreducible over $F$ I've been trying to understand this for a while now, but I'm ...
0
votes
2answers
44 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
2
votes
0answers
38 views

Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
1
vote
1answer
41 views

Is $f/\gcd(f,f')$ always separable?

Suppose $f$ is a nonzero polynomial over an arbitrary field. If $g=\gcd(f,f')$, is it true that $f/g$ is always separable? I was trying to show $\gcd(f/g,(f/g)')=1$. If $d$ is a common divisor of ...
0
votes
6answers
42 views

Prove field extension is a field

I have a field extension $\mathbb Q (2^{1/3}) = a + b2^{1/3} + c2^{2/3}$ where $a,b,c\in \mathbb Q$. I want an elementary proof it indeed is a field. How to go about proving it contains its inverse ...
0
votes
1answer
30 views

How we can prove the irreducibility of polynomials [closed]

Suppose $A,B$ are algebraic over $F$ with minimal polynomial $f$ and $g$ respectively. Prove that $f$ is irreducible over $F(B)$ iff $g$ is irreducible over $F(A)$.
-1
votes
2answers
43 views

What does the idea of splitting mean when used with fields and polynomials?

i want to understand what does field splitting represent,from my book A Course In Galois Theory by D.J.H Garling this term is explained by following sentences Suppose that $K$ is field, that ...
1
vote
0answers
39 views

Factorize polynomial in $\mathbb Z_2[X]$

What is the most efficient way (less time consuming, algorithmically) to factorize polynomials in $\mathbb Z_2[X]$ ? For small degree polynomials, I just try every possibilities (like ...
2
votes
1answer
28 views

A question about the action of $S_n$ on $K[x_1,…,x_n]$

Let ${K}$be the field ($\,Char\,K\not=0)$. Let $n\in \mathbb{Z}^{+}$. $S_n$ acts on $K[x_1,...,x_n]$in the following way: If $p\in K[x_1,...,x_n]$ and $\sigma\in S_n$, then $\sigma p$ is the ...
1
vote
1answer
66 views

About the notation $\mathbb{Z}[x]/(f(x),p)$

Let $f(x)\in \mathbb{Z}[x]$ be a polynomial and $p$ be a prime. What does the notation $\mathbb{Z}[x]/(f(x),p)$ mean? Is it $\mathbb{Z}/p\mathbb{Z}[x]/(f(x))$ ?
1
vote
2answers
54 views

Is this a theorem regarding the solutions of polynomials?

I wanted to refer to this, but I can't remember if this a theorem, named or otherwise, and if it is, how to properly state it. The idea is if we have a solution in radicals to a polynomial with ...
2
votes
1answer
43 views

A field with irreducible polynomial that has multiple roots

Can you give me an example of a field $\mathbb{K}$ such that there exists a polynomial $p(x)\in\mathbb{K}[x]$ that is irreducible and has a multiple root?
1
vote
1answer
28 views

Find a polynomial whose splitting field is $\mathbb{Q}[\alpha,i]$

Let $f(x)=x^{3}-3x+1$ and let $\alpha$ be a root in $f$. i) Show that the polynomial $f$ is irreducible in $\mathbb{Q}[x]$. ii) Show $\alpha^{2}-2$ is a root of $f$ as well, and show that all roots ...
2
votes
0answers
37 views

Field extension of complex root of cubic equation

If $c$ is a complex root of a cubic $a(x)\in\mathbb{Q}[x]$, show that $\mathbb{Q}(c)$ is the splitting field of $a(x)$ over $\mathbb{Q}$. For this, we must show that $\mathbb{Q}(c)$ contains all ...
-1
votes
4answers
77 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.
4
votes
3answers
84 views

minimal polynomial of $\alpha+\alpha^2$ where $\alpha^5=1$ and $\alpha\neq 1$

I have to find the minimal polynomial of of $\alpha+\alpha^2$ where $\alpha^5=1$ and $\alpha\neq 1$ in $\mathbb{Q}$. First I found the minimal polynomial of $\alpha$: $X^5-1=(X-1)(X^4+X^3+X^2+X+1)$ so ...
2
votes
1answer
123 views

Number of distinct roots of an irreducible polynomial divides the degree of the polynomial

Let $f(X)$ be an irreducible polynomial in the polynomial ring $k[X]$ over the field $k$. Prove that the number of distinct roots of $f(X)$ divides the degree of $f(X)$.
5
votes
1answer
67 views

$f(X^p)$ irreducible or $p$th power if $f$ irreducible

An exercise in Bourbaki: Let $K$ be a field of characteristic $p>0$ and $f$ irreducible monic polynomial of $K[X]$. Show that in $K[X]$ the polynomial $f(X^p)$ is either irreducible or the ...
4
votes
1answer
106 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
3
votes
2answers
44 views

minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
7
votes
1answer
82 views

Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
1
vote
1answer
65 views

What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$

I'm doing some exercises to prepare for my exam: What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$. I've no idea how to tackle this ...
1
vote
1answer
70 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
0
votes
2answers
69 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
1
vote
1answer
18 views

Simultaneous irreducibility of minimal polynomials

Let $F$ be a field. Let $u,v$ be elements in an algebraic extension of $F$ with minimal polynomials $f$ and $g$ respectively. Prove that $g$ is irreducible over $F(u)$ if and only if $f$ is ...
7
votes
1answer
74 views

Short method to prove irreduicibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$?

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$? In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...
1
vote
3answers
170 views

Galois group of an irreducible polynomial

Find the Galois group of the polynomial $x^5-9x+3$ over $\mathbb{Q}$. since $3$ cannot divide $a_5$, $3$ divide other coefficients, $3^2$ cannot divide $a_0$, we see that the polynomial is ...
1
vote
2answers
86 views

Finding the minimal polynomial in this field extension of $\mathbb Q$?

I have a field extension $K = \mathbb Q[x]/(x^2 - 5)$ of $\mathbb Q$, and an element $a = \bar x \in K$. I need to find the minimal polynomial of $a$ over $\mathbb Q$. I have worked out that ...
2
votes
1answer
57 views

Polynomials having as roots the sum (respectively, the product) of two algebraic elements

This question remained somehow incompletely solved. The OP also asked for an explicit form of the minimal polynomial of the sum (respectively, product) of two algebraic elements (in a field ...
3
votes
1answer
61 views

Find the order of a polynomial

I want to find the cycle set for the polynomial $p(x)=x^{23}+x^6+1$ over $\mathbb{F}_2$. So, I have the connection polynomial $C(D)=1+D^{17}+D^{23}$ over $\mathbb{F}_2$ The factors to $C(D)$ are: ...
3
votes
1answer
67 views

polynomials factorization over rings and finite fields

Any nonzero polynomial over a subring $R$ of $\mathbb{C}$ is a product of irreducible polynomials over $R$. And for any subfield $K$ of $\mathbb{C}$, factorization of polynomials over $K$ into ...
1
vote
0answers
37 views

Independent indeterminate roots and coefficients of a polynomial

Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of ...
4
votes
1answer
278 views

Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is ...
0
votes
2answers
210 views

Prove that this polynomial is irreducible over $\mathbb Z$

I want to prove that the following polynomial is irreducible: $$x^3 - x^2 - x + 3$$ My question gives the hint to apply the substitution $x \mapsto x+1$ but I've tried this and when multiplied out I'm ...
1
vote
1answer
90 views

On irreducible polynomial over normal extension

Let $L/K$ be a normal extension and a irreducible polynomial $f(X) \in K[X]$. Prove that, if $f$ is reducible over $L$ then $f$ is factored into product of irreducible factors with same degree. ...
0
votes
1answer
161 views

Factor into a product of irreducible polynomials

Since the polynomial $p=x^4−2$ is irreducible over $\mathbb{Q}$, the factor ring $K=\mathbb{Q}[x]/(p)$ is a field. I'd like to factor the polynomial $q=y^4−2$ in $K[y]$ into a product of irreducible ...
1
vote
2answers
84 views

On polynomial of prime degree.

Let $K$ be a field, $f(X)\in K[X]$ be a polynomial of prime degree. Assume that for all extension $L$ of $K$, if $f$ has roots in $L$ then $f$ splits over $L$. Prove that either $f$ is irreducible ...
4
votes
3answers
121 views

$x^5-1$ completely splits in $\mathbb F_{16}$

I need to prove that $x^5-1$ completely splits in $\mathbb F_{16}$. This means it has exactly $5$ unique roots in $\mathbb F_{16}$. I have only found the following way: find an irreducible polynomial ...
0
votes
1answer
62 views

Find the multiplicative inverse of $\,x^2+(x^3-x+2)$ in the quotient $\,F_3[x]/(x^3-x+2)$

Find the multiplicative inverse of $x^2+(x^3-x+2)$ in the quotient $F_3[x]/(x^3-x+2)$ . I've proved that $x^3-x+2$ is irreducible polynomial in $F_3[x]$, and that $x^2$ and $x^3-x+2$ are coprime ...
1
vote
1answer
57 views

Fields of polynomials . Proving that a belongs to k as a root

if $f(x)\in k[x]$, where $k$ is a field, then $a\in k$ is a root of $f(x)$ iff $x-a$ divides $f(x)$ in $k[x]$. My result ... If $a$ is a root of $f(x)=q(x)(x-q)$ and if we let $f(x)=q(x)(x-a)$,then ...
1
vote
1answer
77 views

Automorphisms and Splitting Fields

Note: This question comes from a non-examined question sheet from an undergrad maths course. I want to find the splitting fields of the following polynomials: $x^3-1$ over $\mathbb{Q}$ $x^3-2$ over ...