Finite fields are structures arising in abstract algebra. They are parametrized by their cardinality $q$ that must be a power of a prime number. Notation: $GF(q)$, $\mathbb{F}_q$.
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95 views
Can extending a finite ground field make modules isomorphic?
$\def\Hom{\mathrm{Hom}}$Let $k$ be a field, $A$ a $k$-algebra and let $M$ and $N$ be $A$-modules, finite dimensional over $k$. Let $K$ be an extension of $k$, so $A \otimes K$ is a $K$-algebra and $M ...
7
votes
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100 views
Error correction code handling deletions and insertions
I have data which is expressed in form of fixed-length sequence of decimal digits, typically 10 digits in length.
I'm looking for error correction code that would allow me to append one or more ...
5
votes
0answers
90 views
Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$
Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$.
Here are some examples:
$t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$
$t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
5
votes
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48 views
Subgroup Structure of $\mathrm{SL}(2, p^2)$, and Its Irreducible Characters
I am taking a course in representation theory of finite groups,and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
4
votes
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69 views
Minimal polynomial of a finite purely inseparable field extension
Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
4
votes
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92 views
Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$
I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$.
Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines.
There are $q+1$ ...
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52 views
Deligne and the four Weil statements about polynomials over finite fields?
This article mentions he recently won 3 million of some money. Does anyone have a link to the original Deligne paper that proves this, and could anyone give a basic rundown of how "counting points on ...
3
votes
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109 views
when do two rational elliptic curves have identical size when reduced mod $p$ for all primes $p$?
If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and ...
3
votes
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67 views
Does composing the Frobenius with an automorphism give another Frobenius
Let $X_0$ be a variety over $\mathbf{F}_q$. Consider the Frobenius $F_0:X_0\to X_0$. Let $X= X_0\times \bar{\mathbf{F}_q}$ and let $F:X\to X$ be $F_0 \times \textrm{id}$.
Let $f:X\to X$ be an ...
3
votes
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293 views
Hardness of finding eigenvalues over finite fields
How hard is it (computationally) to find eigenvalues/eigenvectors of matrices over finite fields? Suppose the field has size exponential in the input. (Does the QR algorithm still converge?)
How ...
2
votes
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29 views
Discrete logarithm - strange polynomials
If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number ...
2
votes
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43 views
Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler
I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense:
Each element of $H$ can be represented by one or a few elements of ...
2
votes
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39 views
Different elliptic curves over given $\mathbb{F}_q$ can have different orders?
As the title says: For a given $q \in \mathbb{N}$, in $\mathbb{F}_q$, is it true that different curves on it can have different group orders?
I assume yes. Let $q:=5$. Wikipedia gives $9$ points for ...
2
votes
0answers
87 views
What's the fastest way to solve these equations with powers in a field?
This is for an algorithm I'm working on. Perhaps we can work together!
We can consider the integers modulo a prime $p$. They form a field with arithmetic operations modulo $p$. I'd like to find ...
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50 views
Full Rank Matrix with a specific construction
Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$
$$Z=\begin{bmatrix}
w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
