Questions tagged [finite-fields]
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
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On primitive roots of the form $5^k+10^m$ with $k$ and $m$ nonnegative integers
Let $p$ be any prime. It is well known that the set
$$G_p=\{0<g<p:\ g\ \text{is a primitive root modulo}\ p\}$$
has cardinality $\varphi(p-1)$, where $\varphi$ is Euler's totient function. It is ...
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Polynomial generated with primitive element modulo p
This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let $...
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Dimensions of two spaces
Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.
Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\...
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Distinguish between odd and even powers of a in a finite field
In a finite field $\text{GF}(2^n)$ with characteristic 2, is there a relatively simple way to distinguish between the odd and even powers of the primitive element $a$?
In other words, given an ...
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Eigenvalues of the Cayley-like graph
Let $ F_q $ be a finite field of characteristic 2.
Let $ x^2 + Sx +P \in F_q[x] $ be an irreducible polynomial over $ F_q $,
and let $ g $ be one of its roots in $ F_{q^2} $.
Define a map $ M: F_{q^2}...
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In a finite field with characteristic 2, can I calculate the log(K+1) based on the log(K)?
In the equation $ a^x = a^y + 1$ over a finite field $\text{GF}(2^n)$, where '$a$' is the primitive element, can one calculate $x$ as a function of $y$ without having to resort to taking the logarithm ...
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Probability of summing products of irreducible polynomials in a finite field to zero
Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...
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Vanishing of motivic cohomology with finite coefficients in negative degrees
I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field $\...
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nontrivial cube root of unity [closed]
Hi,
I have a finite field Fp with p = 11 mod(12) and I am trying to get the third nontrivial root of unity in Fp^2 = Fp^2[x]/(x^2+1). So, i need x where x^3=1.
Somehow I came into a source saying ...
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elementary question on ECDLP
If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in ...
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Functions of several variables over finite fields [closed]
For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...
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Bounds for the number of points on projective hyperelliptic curves over finite fields
Let $C$ be projective hyperelliptic curve over finite field $K$.
What are bounds for the number of points $\#C(K)$?
The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are
not smooth ...
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Is there any way to solve this equation without knowing the inverse modulo? [closed]
Suppose I have an arbitrary 256 bit number $m$ another secret number $k$ of the same bit length, and then I multiply them both modulo a 256 bit prime number $p$ to get $c$ as follows:
$$
c = (m\cdot k)...
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Why is there in theory no morphism/isogenies when enlarging a prime field sharing a common suborder/subgroup? [closed]
Simple question : I have a prime field having modulus $p$ where $p−1$ contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as its suborder/subgroup. Why are there no ...
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