# Tagged Questions

42 views

### Silly number theory questions I can't prove.

I know if $gcd(r,s)=1$ then $1=as+bs$ for some intgers $a,b$. Here's what I want to know: which numbers can be written as $as+bs$, if I am restricted to $a,b \in \mathbb{N}$? To be more specific, I ...
25 views

### Multiplicity of roots in finite fields of order prime. [closed]

I am having trouble with completing this question from last years exam (part a and d) Let p be a prime, and $f = x^5-1 = (X-1)(X^4+X^3+X^2+X+1) \in \mathbb{F}_p[X]$ Show: (a) if $p\neq 5$ then every ...
26 views

### Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
43 views

### Polynomials - getting wrong answer using Euclidean algorithm

I am finding the GCD of $a = x^3 + 11/3x^2 + 17/4x + 3/2$ and $b = 3x^2 + 22/3x + 17/4$ using the Euclidean algorithm. So I divide $a/b$ and get $q$ and $r$ such that $a = qb + r$. Then, according to ...
50 views

### Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced polynomial rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of ...
77 views

### Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
31 views

### Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greatest common divisor is 1) then $f(x)$ does not have multiple roots in $K$

Please I would like you to tell me if my proof is correct Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greates common divisor is 1) then $f(x)$ does not have multiple roots in $K$ ...
26 views

### $(x+b)^3\mid P(x)+a$ and $(x-a)^3\mid P(x)-a$

$a,b\in\mathbb{C}$, $b!=0$ I need to find all the polynomials $P$ of degree $5$ verifying: $\begin{cases}(x+b)^3\mid P(x)+a\\ (x-b)^3\mid P(x)-a\end{cases}$ PS : there was en error, i fixed it ...
34 views

### GCD of this polynomial

So here is the exact question that i am having trouble on: "Extend the Euclidean algorithm to polynomials and find the greatest common divisor of: $3x^5-10x^4-4x^3-14x^2-7x-4$ and ...
234 views

### A polynomial with integer coefficient

I'm struggling with this question: Suppose $P(x)$ is a polynomial with integer coefficients such that non of the values $P(1),...,P(2010)$ is divisible by $2010$. Prove that $P(n)\neq 0$ for all ...
91 views

### Divisor in $\mathbb{C}[X]$ $\implies$ divisor in $\mathbb{R}[X]$?

let $P \in \mathbb{R}[X]$ be a real polynomial divisible by a polynomial $Q \in \mathbb{R}[X]$ in $\mathbb{C}[X]$. How can I easily show that $P$ is also divisible by $Q$ in $\mathbb{R}[X]$? A simple ...
42 views

### Euclidean algorithm in the ring of polynomials over a field

I need some help with the following division proofs. I suppose my biggest problem is not being able to visualize the algebra for one GCD equaling another GCD. I'm not sure of how to arrange the ...
97 views

### Show that $(x + 1)^{2n + 1} + x^{n + 2}$ can be divided by x^2 + x + 1 without remainder

I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as P = Q*L + R I am unable to ...
40 views

### An exercise regarding polynomials

I guess it is a simple exercise though I'm not very good at polynomials. It asks: Find $m,n,p,q$ natural numbers such that the polynomial $X^m+X^n+X^p+X^q$ is divisible by $x^3+x^2+x+1$. Thank you in ...
42 views

70 views

### Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $(\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
48 views

### Finding a gcd in the form of a monic polynomial

The question is to find the greatest common divisor (in the form of monic polynomial) in $\Bbb F_5[X]$ of $f=x^2-x+4$ and $g=x^3+2x^2+3x+2$ I used the Euclidian Algorithm for polynomials and found ...
86 views

### Why doesn't this calculation work?

I want to find some closed form for $\gcd(x^3+1,3x^2 + 3x + 1)$ but get $7$ which is not always true.
45 views

### Using long division on polynomials

Can anyone show me how to find $x^5 + 1$ divided by $x^3 + 1$?
39 views

### Find all the polynomials $p \in \mathbb R [X]$ such that $(x+1)p=(p')^2$

(Where $p'(x)$ is the derivative of $p(x)$) Research effort: what I thought is that given that $(x+1)|(p')^2$ then $(x+1)|(p')$ (I'd like to justify better this, but I don't know how) Then, ...
56 views

### If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
61 views

### Simple yet confusing: if $f^2(x)=g^2(x)(x^2+1)$ then $gcd( f^2(x),g^2(x))=(x^2+1)$?

As mentioned in the title: f(x) and g(x) are polynomials above the Rationals field. if $f^2(x)=g^2(x)(x^2+1)$ then does it mean that $gcd( f^2(x),g^2(x))=(x^2+1)$? or maybe it isn't the ...
30 views

### Divisibility of polynomials in $\mathbb{Z}_n[x]$

For what values of $n$ is $x^2+1$ a factor of $x^5+5x+6$ in $\mathbb{Z}_n[x]$? I know how to divide in $\mathbb{Z}[x]$ (with long division), but what should I do here with $\mathbb{Z}_n[x]$, and it's ...
82 views

### Proof polynomial is always divisible by number

Given $f(x) \in \mathbb{Z} [x]$ a polinomyal, that evaluated in any $a \in \mathbb{N}$, results allways in a multiple of 101 or a multiple of 107 (both prime numbers). Prove then, that $f(x)$ it's ...
127 views

37 views

### A question on greatest common divisor

I had this question in the Maths Olympiad today. I couldn't solve the part of the greatest common divisor. Please help me understand how to solve it. The question was this: Let $P(x)=x^3+ax^2+b$ and ...
65 views

### Show that the matrix is invertible

let $A \in M_n(F)$ be a n by n matrix with values from an unknown field $F$. $P_A(t)$ is the characteristic polynomial of $A$, and $g(t) \in F[t]$ a polynomial of an unknown degree. assume that ...
283 views

### Prove $x^n-1$ is divisible by $x-1$ by induction

Prove that for all natural number $x$ and $n$, $x^n - 1$ is divisible by $x-1$. So here's my thoughts: it is true for $n=1$, then I want to prove that it is also true for $n-1$ then I use long ...
39 views

### polynomial division, gcd, question

We are asked to show that there are polynomials $p,q \in Q[t]$ such that: $p(t)*(t^4+2t^2+1)+q(t)*(t^4-3t^2-4) = t^2+1$ Is the answer the same for $t+5$ instead of $t^2+1$? What I tried doing: I ...
83 views

### Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$

Problem: Prove that $x^4+x^3+x^2+x+1$ divides $x^{4n}+x^{3n}+x^{2n}+x^n+1$ for all positive $n$ that are not multiples of $5$. I'd like to get some pointers about how to solve this. No full ...
245 views

### Prove that the Euclidean algorithm for gcd works with polynomials

Given the algorithm $E$: ...
155 views

### Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
62 views