# Questions tagged [polynomials]

This tag is used for both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring and checking for irreducibility.

17,629 questions
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### Irreducible polynomials for that each output is divisible by an integer n

Feel free to delete this question if it has been asked somewhere else before. I've recently stumbled upon this question on the Mathematics StackExchange and I've wondered how the polynomials for ...
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### Is it possible to graph complex zeros of a polynomial?

I am sorry if this question is a complete nonsense, but keep in mind that I am a senior in high school, so my math knowledge is really low. My question is, can you graph complex zeroes on a three ...
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### Tough test polynomials for (finite precision) complex root finding methods, especially Aberth's method

Today I have implemented Aberth's method for complex polynomial root finding. And I have to say I am enchanted about its astonishing performance and its intriguing simplicity. Before I go on believing ...
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### Show that $\frac{x(x-1) \dots (x-n+1)}{n!} \in \mathbb{Z}$ with $x \in \mathbb{Z}$ [duplicate]

Problem: Let polynomial $Q_n (x) = \frac{x(x-1) \dots (x-n+1)}{n!} \in R[x]$ for some ring $R$. Show that $\forall t \in \mathbb{Z}, Q_n (t) \in \mathbb{Z}$. My solution: For each $t \in \mathbb{Z}$, ...
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### If $F\subseteq K$ are fields, $\alpha \in K$ Prove $\alpha$ is algebraic over $F$

If $F\subseteq K$ are fields, $\alpha \in K$, and $K$ is an extension field of $F$. Prove the following are equivalent: $\alpha$ is algebraic over $F$ $F(\alpha)=F[\alpha]$ $|F(\alpha):F|$ is ...
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### Show that any complex polynomial of degree $n \ge 2$ has finitely many neutral or attracting orbits

Let $f: \mathbb{C} \to \mathbb{C}$ be a complex polynomial of degree $n \ge 2$. A point $w \in \mathbb{C}$ is a periodic point of $f$ of minimal period $p$ if $f^{\circ p}(w) =w$ and $p$ is the ...
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### Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...