# Tagged Questions

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### Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
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### Monic polynomials in $(\mathbb Z/p\mathbb Z)[X]$ [duplicate]

Be $p$ a prime number. how many monic polynomials degree 2 there are in $(\mathbb Z/p\mathbb Z)[X]$. How many of them are reducible and how many are not? What I tried to do is, let $f(x)$ be ...
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### $\mathbb{Q}$ adjoining primes and the sum of root of those primes

I have $p$, $q$ as primes, and I want to show that $\mathbb{Q}(\sqrt{p},\sqrt{q})=\mathbb{Q}(\sqrt{p}+\sqrt{q})$. I was thinking about using inclusion both ways, so what does an element in ...
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### Finding a least common multiple (LCM)

My Algebra 2 book explains how to find a least common multiple: Find the least common multiple of $4x^2 - 16$ and $6x^2 - 24x + 24$. Solution Step 1 Factor each polynomial. Write ...
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### Polynomial is prime when evaluated at prime numbers

Let $P(x)$ be a polynomial with integer coefficients such that $P(p)$ is prime for all prime $p$. What are all possible polynomials $P(x)$? Certainly $P(x)=x$ and $P(x)=p$ with $p$ prime satisfy that ...
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### Show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$

Let $n$ be an integer and show that $q(n)=11n^2 + 32n$ is a prime number for two integer values of $n$, and is composite for all other integer values of $n$.
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### Partitioning polynomials in $\mathbb{Z}[x,y]$ by the primes they represent

Suppose you have a set $S\subset\mathbb{Z}[x,y].$ How can one efficiently partition the polynomials into sets such that the primes represented by the polynomials in any given set are identical? For ...
### What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$?
Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value ...