How can I prove that $x^5 + 7x^4 + 2x^3 + 6x^2 – x + 8$ is irrudicible in $\mathbb{Q}$? I can’t use the Eisenstein’s criterion and I tryed to put this polynomial in $\mathbb{Z}_3$ and $\mathbb{Z}_5$. Can you give me some advice please?

Prove that $x^5 + x^2 + 1$ is a primitive polynomial over ${\mathbb F}_2$. I have already proved that the above polynomial is irreducible. Do I have to exhaustively prove that the above polynomial does not divide $X^n + 1$ where $1 \le n < 31$ or is there a better way to prove this? […]

Consider the polynomial $$p(x)=x^9+18x^8+132x^7+501x^6+1011x^5+933x^4+269x^3+906x^2+2529x+1733$$ Is there a way to prove irreducubility of $p(x)$ in $\mathbb{Q}[x]$ different from asking to PARI/GP?

Isn’t $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, it’s irreducible in $\mathbb Z[x]$. Hence, shouldn’t $\mathbb Z[x]/\langle x^2+1 \rangle$ be a field and hence, $ x^2+1$ a maximal ideal in […]

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?

I want to know how to factorize polynomials in $\ GF(2) $ without a calculator in a product of irreducible factors. For example, in my exercises, I must factorize $\ p(x) = (x^7 – 1)$ The response is $\ (x − 1)(x^3 + x^2 + 1)(x^3 + x + 1)$ I just not understand the […]

Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied: 1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ is irreducible. 2) For every $y_0\in\mathbb{Q}$, the polynomial $f(x,y_0)\in\mathbb{Q}[x]$ is irreducible. My question is this: Can we conclude from these two conditions that $f(x,y)$ is an irreducible polynomial in $\mathbb{Q}[x,y]$? I am […]

I have got one of degree 5: $x^5+x^2+1$, but I need one of degree 8.

I have seen the thread Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$ but this didn’t really have a full solution. Is it true that if it is reducible then it can be factored into a linear factor or quadratic factor in the form $x^2 – a$. Is $a$ in the rationals? And what exactly is […]

Prove that $xy-zw$ is an irreducible element in the polynomial ring $\mathbb C[x,y,z,w]$. My attempt was: Consider the homomorphism from $\mathbb C[x,y,z,w]$ to $\mathbb C$ induced by the map $x,y,z,w$ onto $1,2,1,2$ respectively. Since $\mathbb C$ is an integral domain the ideal generated by $xy-zw$ is prime and hence irreducible as $\mathbb C[x,y,z,w]$ is an […]

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