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### 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 ...
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 ...
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### 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.
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 ...
556 views