# Questions tagged [irreducible-polynomials]

Often called prime polynomials. Polynomials that have no polynomial divisors.

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### Is $x^6 + bx^3 + b^2$ irreducible?

Let $b\in \mathbb{Q}^*$ be rational number. We factorise $x^9-b^3\in \mathbb{Q}[x]$ and obtain $$x^9-b^3=(x^3-b)(x^6+bx^3+b^2).$$ Is the polynomial $x^6+bx^3+b^2$ irreducible? If $b=1$ we get a ...
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### Find the product of all irreducible polynomials over $\mathbb{F}_p$ of degree $n$

Let $p$ be a prime number and $n \in \mathbb{N}^+$. Let $H_n$ be the product of all monic irreducible polynomials in $\mathbb{F}_p[T]$ whose degree is equal to $n$. What is known about $H_n$? Is there ...
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Let $f(X)=a_{2n+1}X^{2n+1}+\ldots+a_0\in \mathbb{Z}[X]$ with \begin{align*} a_{2n+1}&\not \equiv 0 \pmod p\\ a_{2n},\ldots,a_{n+1} &\equiv 0 \pmod p\\ a_n,\ldots,a_0&\equiv 0 \pmod{p^2} ... 1 vote 0 answers 71 views ### Reduce Polynomial Over Real Numbers I was given the question x^8 + 16 and told to reduce it as much as able over the real numbers. Here is what I tried x^8 + 16 (x^4+4)^2-8x^4 (x^4+4-2^{3/2}x^2)(x^4+4+2^{3/2}x^2) I can not ... -1 votes 0 answers 48 views ### f(x,y,z) := y^2z + yz^2-x^3+xz^2 is irrreducible in \mathbb{Z}[x,y,z]? Let f(x,y,z) := y^2z + yz^2-x^3+xz^2. Then f is irreducible in \mathbb{Z}[x,y,z] so that I:=(f) is a prime ideal of \mathbb{Z}[x,y,z]? I think that this is my first time of seeing a problem-... -1 votes 1 answer 46 views ### Let L be the splitting field of the polynomial  f over K, Prove that if n!=[L:K], the polynomial is irreducible [closed] Let K be a field and f \in K[x] be a non zero polynomial of degree n. Let L be the splitting field of f over K. Prove that [L:K] divides n! - I already proved this. Now I am stuck at ... 0 votes 0 answers 19 views ### How to generate all elements of an extension field from a base field? (GF(2)) I am trying to understand how to show something is a primitive polynomial; I understand it has to be irreducible by definition, and according to Wolfram: ... 4 votes 2 answers 66 views ### y^{q-1}-f(x) irreducible over \overline{F_{q}}[x] When (\deg f,q-1)=1 I am trying to solve the following problem, and I’d like to ask for some help. Let q be a prime power, f(x)\in F_{q}[x] a polynomial of degree d, such that (d,q-1)=1. I’d like to show that G(... 3 votes 1 answer 66 views ### Are irreducible elements of the tensor product of a vector space equivalent to irreducible polynomials? I'm looking for some feedback on a construction I came across. Loosely, it entails sending a vector space isomorphically to a vector space of polynomials, 'restoring' the ring structure, and asking ... 5 votes 0 answers 134 views ### Is Every Closed Algebraic Set of Dimension n Contained in a Closed Variety of Dimension n+1 Let V be an algebraic variety of dimension m over an algebraically-closed field of characteristic 0, and let n<m and U\subset V be a closed subset of V. Must there exist a subvariety U\... 2 votes 1 answer 69 views ### Maximal number of multiple points for an irreducible quartic I was working on this problem, but I don't see how I can solve it. I was given a hint, but I don't know how to use it. Can anyone help me? Thanks in advance! Let f \in \Bbb C[x_0, x_1, x_2] be an ... 0 votes 0 answers 43 views ### How do I prove that the primitive element of a field extension are this way. I'm doing an introductory course of field theory and there is one excercise that as easy as it seems it bring me on my nerves. It states: Let α_1, \dots, α_n ∈ \mathbb{C} be the roots of an ... 1 vote 1 answer 56 views ### p a prime satisfying p \equiv 3 \mod 4 . Then, the quotient field  F_p [x] / (x^2 + 1) contains \bar{x} that is a square root of -1 I know that x^2 + 1 is irreducible in F_p[x] if and only if -1 is not a square in F_p. Otherwise, x^2 + 1 could be factored out. -1 not being a quadratic residue in F_p is equivalent to ... 1 vote 1 answer 52 views ### On Unique factorization of polynomials I'm studying Lang's Linear Algebra and stumbled upon a lemma prior to the unique factorization of polynomials that says the following "Let p be irreducible in K[t]. Let f, g be non-zero ... 1 vote 1 answer 73 views ### Show that f(x)=x^2+2x-1 \in \mathbb{Z}_3[x] is irreducible over \mathbb{Z}_3. And find the elements of a finite field with 9 elements. Show that f(x)=x^2+2x-1 \in \mathbb{Z}_3[x] is irreducible over \mathbb{Z}_3. Using this fact construct a finite field \mathbb{F}_9 of 9 elements. If \alpha is a root of f(x), then find ... 0 votes 1 answer 50 views ### Generalization or criteria for a proposition for checking irreducibility of polynomials with summand of only two degrees Is the follow proposition Prop Let f \in k\left[x_1, x_2, \ldots, x_n\right] be a polynomial with the form l+h, where l is an irreducible non constant homogeneous polynomial and h is a ... 0 votes 1 answer 100 views ### Question about the solution of the polynomial (x−1)(x−2)⋯(x−n)−1 is irreducible in \mathbb{Z}\left [ x \right ] for all n≥1 The solution of the polynomial (x−1)(x−2)⋯(x−n)−1 is irreducible in \mathbb{Z}\left [ x \right ] for all n≥1 is in here. I think it's no problem that do the same thing on \mathbb{Q\left [ x \... 0 votes 1 answer 40 views ### Reducibility of constrained polynomial Let f \in \mathbb{Z}[x, y] be a polynomial. Suppose that the list of terms of f do not involve the y variable except for a single y^2 term with some arbitrary coefficient. When is f ... 2 votes 2 answers 155 views ### Quintic equation with integer coefficients I am looking for a way to find a closed form of the real root of the quintic eq. with integer coefficients: x^5+3x^4+4x^3+x-1=0. According to the numerical calculation the root x_0\approx 0.... 1 vote 1 answer 37 views ### Is there a special name for linear irreducible polynomials (over the complex numbers)? According to the fundamental theorem of algebra every polynomial over the complex numbers can be factorized into the following form: c (x - r_1) (x - r_2) (x - r_3) \dots $$where r_i are the ... 0 votes 1 answer 59 views ### x^m+y^m+1 is irreducible in k[x,y] Question: Let k be a field with characteristic 0. Let m\geq 2 be an integer. Show that f(x,y)=x^m+y^m+1 is irreducible in k[x,y]. Answer: I have no idea how to solve this question. Any hint/... 1 vote 1 answer 40 views ### Let \alpha \in \mathbb{C} a root of f(x)=x^3-3x-1. Prove f is irreducible over \mathbb{Q}. I encountered the following claim: Let \alpha \in \mathbb{C} a root of f(x)=x^3-3x-1. Prove f is irreducible over \mathbb{Q}. The explanation included something of the form: Since f has ... 3 votes 0 answers 62 views ### Different approach to see that X^{nm}-2^n3^m is irreducible over \mathbb{Z} [duplicate] I came across the problem to show that when n,m\in \mathbb{N} are coprime then the polynomial X^{nm}-2^n3^m is irreducible over \mathbb{Z}. I solved it appealing to knowledge of complex numbers, ... 0 votes 0 answers 60 views ### p odd prime and n integer. Prove that x^n - p is irreducible over \mathbb{Z}[i] By Eisenstein's criterion I know that the polynomial is irreducible in \mathbb{Z}[X], since p divides p and p^2 does not divide p. But I do not know how to extend to \mathbb{Z}[i]. 0 votes 1 answer 98 views ### What are elements of the field \frac{\mathbb{F}_2[x,y]}{x^3-y^2+x+1}? Is that polynomial irreducible? On a previous problem, I had the field A=\mathbb{F}_3[x] and the polynomial p(x)=x^3+x+1, where that polynomial is reducible in \mathbb{F}_3, then I had \frac{A}{p(x)} and I had to find its ... 0 votes 1 answer 60 views ### Prove that f is irreducible if it has no roots in a finite field F. [closed] Let f\in\mathbb{Z}[X] be a monic polynomial with \text{deg}(f)=5. Suppose that there exist a prime number p and a finite field F of order p^2 such that f has no roots in F. Prove that f... 3 votes 1 answer 96 views ### How can I know if the polynomial x^4 -16x^3 +12x^2 - 3x + 9 is irreducible over \mathbb{Z}? How can I know if the polynomial x^4 -16x^3 +12x^2 - 3x + 9 is irreducible over \mathbb{Z}? I have tried to use Eisenstein's criterion by evaluating on polynomials of the form ax+b but I have not ... 6 votes 1 answer 92 views ### Does \sqrt a + \sqrt b have a four way conjugate? Let a, b be rational numbers that are not perfect squares. Consider the set S = \{\sqrt a + \sqrt b, \sqrt a - \sqrt b, - \sqrt a + \sqrt b, -\sqrt a - \sqrt b\}. If p is a polynomial with ... 3 votes 2 answers 194 views ### x^6 + 69x^5 − 511x + 363 is irreducible over \mathbb Z? As mentioned, I am trying to show that x^6 + 69x^5 − 511x + 363 is irreducible over \mathbb Z. To see that it has no roots and no cubic factors, I send the polynomial to \mathbb F_7 and \mathbb ... 1 vote 1 answer 36 views ### Knowing that f' has a rational root, what can we say about the discriminant of a root of f, where f is monic and irreducible over \mathbb{Z} Let f(x) be a monic irreducible polynomial over \mathbb{Z} and let \alpha be a root of f. Then I have to show that f(r)~|~disc(\alpha), when f'(x) has a root r in \mathbb{Z}. These are ... 2 votes 1 answer 40 views ### Does non-zero abolute trace of an element \alpha imply the irreducibility of f(x)=x^p-x-\alpha I am currently reading the paper "Fast Contruction of Irreducible Polynomials over Finite Fields" by Couveignes and Lercier. On page 81, it reads, "... So 1/(1-b) is a root of the ... 1 vote 1 answer 61 views ### Monic irreducible polynomial f \in \mathbb{Z}[x] of degree n such that \operatorname{Gal}(f) \cap C_j \neq \emptyset  \forall j… I’m trying to solve the following problem: Let C_1, \ldots, C_m \subseteq S_n be conjugacy classes of elements in S_n. Show that there exists a monic irreducible polynomial f \in \mathbb{Z}[x] ... 2 votes 1 answer 117 views ### -3x^{2m}+7x^m-3 is irreducible for all m\geq 1 I have heard that p(x)=-3x^2+7x-3 is the simplest polynomial for which p(1)=1, p(x)=x^{\deg p}\cdot p(x^{-1}) and p(x^m) is irreducible for all m\geq 1. I have tried to show the last part, i.... 1 vote 0 answers 76 views ### Proving that a smooth affine variety is irreducible I am struggling with the following problem. Given a complex polynomial f : \mathbb{C}^n \rightarrow \mathbb{C} such that \nabla f doesn't vanish anywhere on the whole \mathbb{C}^n, V = V(f) is ... 1 vote 1 answer 72 views ### Is x^{100} - x^2 + 1 separable in an algebraic closure of \mathbb{F}_2 My approach: f'(X) = 100x^{99} - 2x = 0x^{99} - 0x = 0 since in \mathbb{F}_2. So the \gcd(f,f') = f > 1, thus not separable. On the other hand, f(0) \neq 0 \neq f(1), so irreducible. But ... 1 vote 1 answer 40 views ### Multiple roots of irreducible polynomials over fields of positive characteristic It can be proved that if K a field, then f \in K[x] has a as a multiple root if and only if f(a) = f'(a) = 0. And as a corollary, if K has characteristic 0, then irreducible polynomials do ... 0 votes 1 answer 54 views ### Is 13x^5 + (3 − i)x^3 + (8 − i)(x^2 − x) + 1 − 2i irreducible in (\mathbb{Q}[i])[x]? Is$$13x^5 + (3 − i)x^3 + (8 − i)(x^2 − x) + 1 − 2i$$irreducible in (\mathbb{Q}[i])[x]? I've tried using Eisenstein’s irreducibility criterion to prove that it is, but I don't think it applies ... 2 votes 1 answer 67 views ### Show that \langle X_1X_4-X_2X_3 \rangle is irreducible in \mathbb{Q}[X_1,X_2,X_3,X_4]. In lecture we did the following example. Show that X_1X_4-X_2X_3 is irreducible in \mathbb{Q}[X_1,X_2,X_3,X_4]. We wrote down that if X_1X_4-X_2X_3 = a\cdot b for some a,b \in \mathbb{Q}[X_1,... 1 vote 2 answers 67 views ### Factorization over \mathbb{Q} and \mathbb{Z_{41}} Factor f(x) = x^4+1 over \mathbb{Q} and over \mathbb{Z_{41}}. 1)I can't factor f(x) over \mathbb{Q} because f(x+1) is irreducible by Eisenstein's criterion. 2)I don't know where to start: ... 3 votes 2 answers 111 views ### Showing that f=X^p-X+T is irreducible over \mathbb{F}_p(T)[X] Let K=\mathbb{F}_p(T) be the field of rational functions on one variable T over \mathbb{F}_p, and f=X^p-X+T \in K[X]. I want to show that f is an irreducible polynomial. I know that T is a ... 0 votes 2 answers 45 views ### x^{\frac{p-1}{2}}+1 is reducible in \Bbb Z_p[x] [closed] Let p be an odd prime. Prove that the polynomial f(x) = x^{\frac{p-1}{2}}+1 is reducible in \Bbb Z_p[x] and factor f(x) into irreducible polynomials in \Bbb Z_p[x]. I've been struggling ... 1 vote 1 answer 42 views ### Dimension of \mathbb{Q(\omega)} and minimal polynomial of \sqrt{2} Consider:$$\omega = \frac{-1}{2} + \frac{\sqrt{3} i}{2}$$and the simple extension \mathbb{Q(\omega)}. Find the dimension of \mathbb{Q(\omega)} and the minimal polynomial of \sqrt{2} over \... 1 vote 0 answers 53 views ### Ask for help on proving irreducible polynomial on K[x] Let F be a field and a,b\in F with a\ne0. Then, f(x)\in F[x] is irreducible if and only if f(ax+b)\in F[x] is irreducible. This is my proof (\Rightarrow) Suppose f(x)=h(x)g(x) is ... 0 votes 0 answers 36 views ### An irreducible polynomial over Zp [duplicate] In general, there is a problem: "Prove that the polynomial f(x) = x^{p} - x - 1 is irreducible over \mathbb{Z}p (p is a prime number)". I had an idea to solve this problem using the ... 0 votes 1 answer 66 views ### Some properties about \mathbb{F}_3[x]/(x^3+x+1) I am given L:=\frac{\mathbb{F}_3[x]}{(x^3+x+1)} and I have to prove different properties about this object. First of all, since the polynomial for which I make the quotient is reductible$$x^3+x+1=(...
Consider the polynomial $$27x^4 - 256 y^3 = k^2,$$ where $k$ is an integer. As $k$ varies over all positive integers, is it possible to show that there are infinitely many distinct integral ...