# Tagged Questions

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### Points on the intersection of an affine quadric and cubic over a finite field

Are there absolute constants $N$ and $B$ such that the following is true? Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with ...
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### How does Tate verify his own conjecture for the Fermat hypersurface?

This question is about Tate's 1963 paper "Algebraic Cycles and Poles of Zeta Functions". Here he announces a conjecture (now known as "the Tate conjecture") which states that certain classes in the ...
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### Irreducibility of some trinomials modulo $p$

Let $n>1$ be an integer. An old result of Selmer, See Theorem 1, page 289 in http://www.mscand.dk/article.php?id=1472, (If the link does not work try googling: ...
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### Quadratic forms without common zeroes

A question which, I suppose, is as easy as abc for an expert. For a given finite field $F$ of odd characteristic (if needed, the characteristic is $3$ and the size of $F$ is large), do there exist two ...
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### Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$: For each positive ...
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### Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?

Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$. Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
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### Existence question on rational points on a curve

I am puzzled about the following question: Let C be a smooth, projective, absolutely irreducible curve defined over GF(q) and let g denote the genus of C. O is a rational point on C, and the divisor ...
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### Number of irreducible polynomials with some coefficients fixed over a finite field

I am interested in the following problem: I have a finite field $F_q$, two positive integers $n>m$ and elements $a_1,...,a_m\in F_q$. How many of the polynomials ...
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This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
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### Number of solutions of a linear equation in a small subset.

Let $p$ be a prime. Let $F_p$ be the finite field of $p$ elements. Let $A$ be a subset of $F_p$ of size $s$. Assume that $s > 2$ is polylogarithmic in $p$. Suppose that we want to count ...
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### Roots of unity in different completions of a number field

For any field $k$, let $\mu(k)$ denote the roots of unity in $k$. Now let $k$ be a number field and let $v, w$ be non-archimedean primes of $k$ with distinct residual characteristics. Does there ...
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### additive structure in a small multiplicative group of a finite field?

Let $p$ be a prime. Given a positive integer $n$, does there exist a $\beta$ in an extension of $F_p$ such that 1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a high extension) ...
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### How to factorize X^n - 1 in Z/pZ?

How do I factorize a polynomial $X^n - 1$ over $\mathbb{F}_p$? In particular I need to find factors of the polynomial $X^{3^3 - 1} - 1 = X^{26} - 1$ over $\mathbb{F}_3$.
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Let $q$ be a power of 2. Let $P$ be the set of polynomials in $F_q [x]$ of degree d or less. Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the number of distinct ...
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### Local-globalism for similar matrices?

My background on number theory is very weak, so please bear with me... Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in ...
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### Counting solutions to x^{p+1}=y^4 in a finite field

I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would ...
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### Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...
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### A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...
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### Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree ...
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### Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with q elements says that: the eigenvalues of ...