Proving C[x,y]/⟨x2+y2+1⟩,R[x,y]/⟨x2+y2+1⟩\mathbb C[x,y]/\langle x^2+y^2+1\rangle,\mathbb R[x,y]/\langle x^2+y^2+1\rangle are integral domains

As a homework assignment, I need to prove that C[x,y]/⟨x2+y2+1⟩ and R[x,y]/⟨x2+y2+1⟩ are integral domains. I have no idea how to approach problems like this. We’re allowed to use the fact that for any field, F[x,y]/⟨xy+b⟩ is an integral domain iff b≠0. Here are some thoughts: If x2+y2+1 is irreducible then it generates a prime … Read more

Can we use Eisenstein’s Irreducibility Criterion to show that x4+1x^4+1 is not reducible in Q?

As such: Let a(x)=x4+1∈Q[x]. Then choose any prime p. By Eisenstein’s Criterion, we see that p∤, p\mid 0 (since all coefficients of intermediate terms are 0), and p^2\nmid 1. Thus we conclude that a(x) is not reducible in \mathbb{Q}. Is this valid, or am I making some glaring omission? My professor used the Rational Root … Read more

Irreducibility of a polynomial with no roots over KK, charK=p\mbox{char} K= p

Let K be a commutative field with characteristic p∈P and the polynomial K[x]∋f(x):xp−x+a,0≠a∈K where f has no roots over K. Show that f is irreducible. Quick side-step: it’s generally incorrect to assume having no roots implies irreducibility. For instance, if we look at x2+1 over R, it has no roots, hence the fourth degree polynomial … Read more

Xpn−XX^{p^n}-X is the product of all irreducible, normalized f∈Fp[X]f \in \mathbb F_p[X]

I’d like to show the following: Xpn−X is the product of all irreducible, normalized f∈Fp[X] with deg(f)∣n in Fp[X] To me, the claim above feels kinda similar to this theorem: An irreducible polynomial f∈Fp[X] is a divisor of Xpn−X iff deg(f(x))=d divides n. But I don’t know how I can connect those two (or is … Read more

Factor x5+x2+1x^5+x^2+1 into irreducible polynomials in Z[x]Z[x]

So here is my question: i would like to determine if whether or not the polynomial x5+x2+1 in Z[x] is irreducible and if not then find the factors. I tried a lot to find it. I really think it is already irreducible but I don’t know how to prove it. I tried Eisenstein’s criteria which … Read more

How to formally prove that the degree of Q(5√6,3√7)\Bbb Q(\sqrt[5]{6},\sqrt[3]{7}) is 1515?

I tried to consider the tower of extension Q⊂Q(3√7)⊂Q(5√6,3√7). The minimal polynomial of Q(3√7) over Q is x3−7 by Eisenstein. But although it is easy to see that it has no root in Q(3√7), how can I formally conclude that x5−6 is irreducible over Q(3√7) and thus we can see the basis of Q(3√7,5√6)? Thanks! … Read more

Sum of roots of unity while computing irreducible polynomial

I’m trying to compute the minimal polynomial of a root of ξ+ξ−1 where ξ is a fourteenth root of unit, that is, x14=1. Using Galois theory I was able to determine that the minimal polynomial is (X−(ξ+ξ−1))(X−(ξ4+ξ−4))(X−(ξ2+ξ−2)) my problem now is that it seems not evident how to expand this product. How does one compute … Read more

For $p$ prime, is the polynomial $x^p-x+1$ irreducible in $\mathbb{Z}_p$? [duplicate]

This question already has answers here: How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? (8 answers) Closed 3 years ago. It is possible, for $p\in\mathbb{N}$ prime, that the polynomial $x^p-x+1$ is irreducible in $\mathbb{Z}_p$? By the identity $a^p\equiv a$ mod $p$ for any $a\in \mathbb{Z}_p$ surely … Read more