All Questions
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Reducible/Irreducible Polynomials in Ring Theory
I have this following exercise I've been trying to solve for a while now.
We are supposed to study the irreducibility of the polynomial $A=X^4 +1$ in $\mathbb{Z}[X]$ and in $\mathbb{Z}/p \mathbb{Z}$ ...
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Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ (Homework) [closed]
Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ This problem is from a homework set of my class at source: Alphastar.academy. I believe there a ...
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Using the remainder theorem to prove a quadratic is a factor of a polynomial
For example, if I have $P(x) = 3x^4 + 5x^3 -17x^2 -13x + 6$ then to show that $x^2 + x - 6$ is a factor I individually show that $x+3$ and $x-2$ are factors using the factor theorem (i.e. $P(-3) = 0$ ...
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Irreducibility of polynomials in $F[X]$
Let $f = 3X^4 + 2X^3Y + X^2Y^2 + 3XY^3 + Y^4$. Let $R =\mathbb{Z}[Y]$, $F = Frac[R]$.
I want to know whether or not $f$ is irreducible in $F[X]$. I've tried using the Eisentstein criterion, but I ...
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How to factorize the expression $4(ab+cd)^2-\left(a^2+b^2-c^2-d^2\right)^2$? [closed]
Factorise$$4(ab+cd)^2-\left(a^2+b^2-c^2-d^2\right)^2$$
I can't solve this math assignment from my text book. No one knows how to solve it, so I would be so thankful to you if you presented your step-...
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An Analogue of CRT for Polynomial Ring
How can we determine the number of factors by CRT?
For example,
$$\frac{\mathbb{Z}_5[X]}{X^2+1}\cong \frac{\mathbb{Z}_5[X]}{X+2}\times \frac{\mathbb{Z}_5[X]}{X+3},$$
so $\frac{\mathbb{Z}_5[X]}{X^2+...
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prime values polynomial confusing question
The number of prime values of the polynomial $n^3 − 10n^2 − 84n + 840 $ where $n$ is an integer is??
I do not get what they're asking us to do. Is there a specific method to solve this question? I ...
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Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$? [closed]
The question is as in the title:
Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
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