Linked Questions

1
vote
1answer
42 views

An Irreducible polynomial in $\mathbb{Q}[x]$ [duplicate]

I want to prove that $x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Q}[x]$. I noticed that this polynomial can be rewritten as $x(x+1)(x^2+1)+1$. As seen, it has no proper divisor because of the ...
0
votes
1answer
89 views

cardinality of Galois group in $\mathbb{Q}$($\zeta_n$) [duplicate]

Let $\mathbb{Q}$($\zeta_n$) be some cyclotomic field, where $\zeta_n$ is a n-th root of unity. I already managed to show that $\mathbb{Q}$($\zeta_n$) is an Galois extension, but now i struggle to ...
0
votes
1answer
46 views

If $\zeta_n$ is a primitive $n$th root of unity, why is $\text{dim}_{\Bbb Q}\Bbb Q[\zeta_n]=\phi(n)$? [duplicate]

I have no idea what cyclotomic polynomials are and how we can get the result using that. Is there another way to prove it? Any hint is appreciated.
11
votes
5answers
3k views

Show that $x^4-x^2+1$ is irreducible over $\mathbb{Q}$

My attempts: I cannot apply the Eisenstein's criteria here, because there is no prime number that divides the constant term i.e. $1$ Taking a translation of the form $x \rightarrow x+a$ does not ...
19
votes
2answers
556 views

Which vectors can give zero inner products forever

For even positive integer $n$, consider an $n$-dimensional vector $v$ such that $v \in \{-1,0,1\}^n$. Now consider an infinite dimensional vector $w$ with $w_i \in \{-1,1\}$ and define $I_k = \sum_{i=...
0
votes
3answers
260 views

Find the degree of extension $\Bbb Q(ζ_9 + ζ^{−1}_ 9 )$ over $\Bbb Q$

I am trying to find the degree of extension $\Bbb Q(ζ_9 + ζ^{−1}_ 9 )$ over $\Bbb Q$. I know that for a prime number $p$, we always have $[\Bbb Q(ζ_p):\Bbb Q]=p-1$, so we can use the tower of ...
3
votes
1answer
596 views

Beating Gauss on Irreducibility of Cyclotomic Polynomials?

I am considering how to provide an alternative proof of the lemma used in the proof that $\Phi_n$ is irreducible: Lemma: If $\Phi_n=f_1 f_2\cdots f_r$ is a factorization into monic irreducible ...
3
votes
2answers
320 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 $12^{\text{...
3
votes
1answer
305 views

Bounding the multiplicative order of matrices in $\mathbb M_n(\mathbb Z)$

Let $\mathbb N$ be the set of positive integers. Prove that: $\forall n\in\mathbb{N}:\exists r\in\mathbb{N}$ (let's say $r=r(n)$ as a function of $n$) such that: If $M\in \mathbb{M}_n(\...
1
vote
1answer
536 views

Automorphism that maps primitive roots of unity.

Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ ...
2
votes
2answers
147 views

$\left[\Bbb{Q}\left(\cos\left({2\pi\over n}\right)+i\sin\left({2\pi\over n}\right)\right):\Bbb{Q}\right]=\phi(n)$

How to prove that $$\left[\Bbb{Q}\left(\cos\left({2\pi\over n}\right)+i\sin\left({2\pi\over n}\right)\right):\Bbb{Q}\right]=\phi(n)$$ Here $[F(a):F] $ is basically the index of field $F(a)$ over $...
4
votes
2answers
229 views

how to find all the solutions to $I+A+\cdots+A^n=0.$

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying $$...
1
vote
2answers
133 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: https://www.maa.org/sites/default/files/...
6
votes
1answer
144 views

Find an example of degree-100 extension of $\Bbb Q(\zeta_5)$ and $\Bbb Q(\sqrt[3]{2})$.

I am trying to find an example of degree-100 extension of $\Bbb Q(\zeta_5)$ and an example of degree-100 extension of $\Bbb Q(\sqrt[3]{2})$. For the example of degree-100 extension of $\Bbb Q(\sqrt[3]...
1
vote
1answer
124 views

Cyclotomic polynomials being irreducible over Q

So, task is to, using algebra, write polynomial $X^n-1$ as a product of irreducible polynomials over $Q$. Our prof told us that the solution is $$X^n-1 = \prod_{d|n} \Phi_d(x),$$ where $\Phi_d(x)$ ...

15 30 50 per page