Questions tagged [finite-fields]
Finite fields are fields (number systems with addition, subtraction, multiplication, and division) with only finitely many elements. They arise in abstract algebra, number theory, and cryptography. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.
4,853
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Number of Solutions of the Hyperbola Equations over Finite Fields
I have a problem with proving the number of points of the hyperbola equation $H_a: x^2 + y^2 = a$ (for every a > 0 in finite field $F_p$) in the finite fields. I have to prove that the number of ...
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Is spliting field of $f(x) \in F[x]$ with $F$ a finite field also a finite field?
Let $F$ be a finite field, and $f(x) \in F[x]$ is some non constant polynomial, Let $E$ be the splitting field of $f(x)$ over $F$ , is $E$ always a finite field?
My attempt let $\deg(f) = n$ then ...
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Multiplication of two random matrices over a finite field
Consider a matrix $\mathrm{X}$ sampled uniformly at random from the set of all rank $r$ matrices over $\mathbb{F}_q^{m \times n}$ and a matrix $\mathrm{Y}$ sampled uniformly at random from the set of ...
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How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]
My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it.
In my second attempt, I tried to look at them as vector ...
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Proving that the following equation does not have integer solutions
I want to prove that the following equation has no integer solutions $a,b,c$: $$-a^3 - b^3 - c^3 + ab^2 - ac^2 + bc^2 - 2a^2c + 3abc = 0$$ apart from the naive solution $a=b=c=0$.
The context, in case ...
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Showing that $x^3 - t$ is irreducible over $\mathbb{F}_3(t)$
I was reading the post Is $\mathbb{F}_3(t,t^{1/3})/\mathbb{F}_3(t)$ a normal extension? Is it separable?
I do not understand, why we can use Eisenstein's criterion to show that $x^3 - t$ is ...
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show $ord(-1)=1$ in $K^*$ where $|K|=p^n$ is a field
Consider a finite field $K$ of order $|K|=p^n$. Denote $K^*$ the group of units.
Show that if $p=2$ then the $ord(-1)=1$ in $K^*$
Show that if $p\geq3$ then $ord(-1)=2$ in $K^*$
How do I approach ...
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A lemma about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra
Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.)
I am confused by ...
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Lemma A-5.19 about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra
Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. It is about the characterization of the Galois group of pure extensions (which are mostly called radical ...
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find all irreducible polynomials of degree 2 and 3 over Z5
I would like to find all irreducible polynomials of degree 2 and 3 with coefficients in Z5.
I know that the polynomial (x^5)^n - x equals with the product of all monic irreducible polynomials of ...
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Over a finite field, which square matrices produce a zero quadratic form?
For which matrices $A \in (\mathbb{F}_p)^{n \times n}$ do we have
$x^T A x=0$ for all $x \in (\mathbb{F}_p)^n$?
Obviously, this is the case if $A=B-B^T$ for some $B$ (which is equivalent to saying ...
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Sum over exponentiated bilinear form in finite-field vector space
Let $A$ be a linear map over the finite-field vector space $(F_2)^n$, i.e., an $F_2$-valued $n\times n$ matrix, not necessarily symmetric. I'm interested in the sum
$$\sum_{X\in F_2^n} (-1)^{X^T A X}\...
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Sum over bilinear form in finite-field vector space
Let $A$ be a linear map over the finite-field vector space $(F_2)^n$, i.e., an $F_2$-valued $n\times n$ matrix, not necessarily symmetric. I'm interested in the sum
$$\sum_{X\in F_2^n} X^T A X\;.$$
...
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x^7-1 over F_3 is not solvable by radicals [closed]
Showing that the polynomial $x^7-1$ over $\mathbb{F}_3$ is not solvable by radicals.
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Reference request for perfection of schemes over finite fields
I am currently reading a paper from 2021 which uses "perfection" of schemes over finite fields. If $X$ is such a scheme over $\mathbb F_q$, the associated perfection is denoted by $X^{\...
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Centralizer/normalizer of degree $n$ extension embedded in $\mathrm{GL}_n(k)$
Let $k$ be a finite field, and let $\ell/k$ be the unique degree $n$ extension of $k$. Then, by choosing a basis for $\ell$, we can identify $\ell^\times$ with a subgroup of $G=\mathrm{GL}_n(k)$. Such ...
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Finite fields, trace map and the induce map is being a permutation of $ \mathbb F_{2^n}$
$n\ge 2$ is an integer
$$tr : \mathbb F_{2^n} \to \mathbb F_{2} \\ tr(x)=x+x^2+x^{2^2}+\cdots x^{2^{n-1}}$$ is the absolute trace map
For fixed constants $\alpha, \theta \in \mathbb F_{2^n} $ define ...
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If $\operatorname{tr}(ab)=0$, then $f(x)=x+a\operatorname{tr}(bx)$ is a permutation
Let $L/K$ be finite fields with elements $a,b\in L$ , with $\operatorname{tr}$ being the trace map and $f:L\to L$ given by $f(x)=x+a\operatorname{tr}(bx)$. Then,is it true that, $\forall x\in L$, if ...
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Hyperplane arrangement : The Shi arrangement
I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial ...
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Order of Affine non-soluble groups
2-Transitive groups has been classified. The complete table has been mentioned in this Textbook ( see e.g., Table 7.3 and Table 7.4, page no. 194-197). Table 7.3 contains Affine 2-transitive groups ...
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Order of group $A\Gamma L_d(F)$ of Affine semi-linear transformation.
Question What is the order of a group $A\Gamma L_d(q)$ of Affine semilinear transformation ? (here $q$ is a prime power of some prime $p$).
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How to solve the $p$th power of matrix $C$?
if $p\equiv1(mod3)$
, $C=\left[
\begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & -3 \\
0 & 1 & -3 \\
\end{array}
\right]\in M_3(\mathbb{F}_p).$
How to calculate $C^p$?
...
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Two meanings of $0$ in field axioms for $\Bbb Z_5\times\Bbb Z_5$?
The additive identity in a field is unique, and we call it $0$.
The field axioms say that every element except $0$ has a multiplicative inverse.
But consider the field $\Bbb Z_5$. It has a $0$ ...
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Solving a system of equation over finite fields with some extra condition
Consider the system of modular equations $$c_{1}f_{1}+c_{2}f_{2}+c_{4}f_{4}+ 2c_{5}f_{5} \equiv 0 \hspace{1mm} (\text{mod }3) \\ c_{1}f_{1}+c_{3}f_{3}+c_{4}f_{4}+ 2c_{5}f_{5} \equiv 0 \hspace{1mm} (\...
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How to factor $X^4+5X^3-2X^2-2$ into its irreducible form over $\Bbb{Z}_{11}$ [closed]
The polynomial $X^4+5X^3-2X^2-2$ has no roots in $\Bbb{Z}_{11}$ so I am unsure as to how I am meant to factorise in such a scenario when I cannot use the factor theorem. How am I meant to progress? ...
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When is the trace of a matrix group surjective over $\mathbb{F}_p$?
Let $p$ be a prime and $G\subset\operatorname{GL}_n(\mathbb{F}_p)$ be a subgroup. I wondering about the following question:
Is the map $\operatorname{Tr}:G\to\mathbb{F}_p$ surjective?
I know it's true ...
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Is it possible to produce identically-behaving binary extension fields using different irreducible polynomials?
Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$.
Is there any possibility that two (or more) different $f(z)$ can ...
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Polynomial in finite field where all the elements of the field are roots
Find a monic polynomial of degree n where n is a power of a prime p and every element of F_n is a root.
I attempted to solve this problem by using the fact that every element in a finite field can be ...
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Vector spaces over certain finite fields are not equal to any unions of their subspaces
From the 3rd edition of the book "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan, we find the following exercise under chapter 3:
:
First of all, let's ...
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Linear algebra over finite fields
Let $\mathbb{F}_p^3$ be a $3$-dimensional vector space over $\mathbb{F}_p$ with $p$ odd. For any $\mathbf{x}\in \mathbb{F}_p^3$ define its "norm" $\lVert \mathbf{x}\rVert=x_1^2+x_2^2+x_3^2,$ ...
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How to construct a cubic monic irreducible polynomial over $\mathbb{F}_p[x]$? $p$ is prime.
I'd like to know that how to construct a cubic monic irreducible polynomial over $\mathbb{F}_p[x], p$ is a prime number.
It is known that the polynomial is irreducible iff it has no roots in $\mathbb{...
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Genus of curves over finite fields
How does one define genus of curves defined over finite fields (or over general fields)?
I only know the geometric definition for smooth plane curves $g = \frac{(d-1)(d-2)}{2}$, and the definition for ...
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Generalization of convolution theorem [closed]
I am for years interested in things related to convolution, and different groups, and fast transformations
If the convolution is defined as
$$\sum_{j+k \equiv i \operatorname{mod}N} u_j v_k$$
Then we ...
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Let $L/K$ be a field extension, s.t $[L:K]>2$, and $\sigma\in Aut(L/K)$ s.t $\sigma$ is of order 2, prove that $L^\sigma\notin\{\mathbb{Q},F_p\}$.
I am given a field extension $L/K$ and an automorphism $\sigma \in Aut(L/K)$ such that we define the field $L^\sigma:=\{ x\in L: \sigma(x)=x\}$, then I want to show that for the case when $\sigma$ is ...
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Computing $\chi(1)$ and $\chi(s)$ for $\chi\in\widehat{\mathrm{GL}_2(\mathbb{F}_q)}$ and semisimple non-regular $s$ using formulas of Deligne-Lusztig
Let $G=\mathrm{GL}_2$ and $s=\left(\begin{smallmatrix} a & \\ & b \end{smallmatrix}\right)$ be semisimple and non-regular in $G(\mathbb{F}_q)=\mathrm{GL}_2(\mathbb{F}_q)$ (i.e. $a\neq b$ and $...
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A question related to the divisibility of a number
I am currently reading upon the following proof of the following statement from Sharifi's notes on Algebraic Number Theory
The maximal unramified extension $K_{nr}$ of a local field $K$ is
given by ...
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DFT, but with large values available
Suppose that we are calculating a size $N$ integer-valued DFT, with some values possibly adjoined to the integers, such as the imaginary $i$. My question is, if the word size allows integers much ...
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Equivalent of floor division in a group of integers mod N.
I'm working on a small programming project, and I'm struggling a bit with calculating fractions of numbers in a commutative group. I'm by no means a mathematician or a programmer, so please bear with ...
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$\exists$ $a, b, \in \mathbb{F}$ such that $a^2 + b^2 = 2ab = 0$ $\implies$ Char($\mathbb{F}$) $= 2$.
Consider a field $\mathbb{F}$.
If there exist elements $a, b \in \mathbb{F}$, not both zero, such that $a^2 + b^2 = 0$ and $2ab = 0$, then I need to show that the characteristic of $\mathbb{F}$ is $2$....
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Inequality involving degrees of field extensions
Suppose we have fields $F \subset E \subset K$ and $\alpha \in K$. Suppose also that $K$ is algebraic over $E$. Let $\alpha \in K$. Then we have $p(\alpha)=0$ for some $p(x) = a_{0} + a_{1}x+ \dots+a_{...
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Possible $|F(\alpha, \beta):F|$ where $|F(\alpha):F|=6$ and $|F(\beta):F|=15$?
Here is a past paper problem which I am struggling to solve currently.
Let $\alpha,\beta\in E$, where $E$ an extension of field $F$. We are given $|F(\alpha):F|=6$ and $|F(\beta):F|=15$.
What are the ...
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Irreducible polynomial in integers modulo p
I am a completing a past paper question and I am undecided on what method to use here. The question is:
For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are:
(1) Check each $a\...
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Show that the units of a finite field form a cyclic group.
I have shown that for a finite commutative group, there is an element $x$ such that the order of every other element divindes the order of $x$. I was thinking how I could apply this to prove that the ...
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Representation theory over finite fields?
This is mostly a reference request ... I think.
I am a bit familiar with representation theory of finite groups. Here I have seen a representation as a homomorphism $\rho: G \to GL(V)$ where $V$ is a ...
- 2,797
2
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1
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Modulo calculation on a polynomial, in NASA tutorial on Reed-Solomon codes
I am reading Geisel's tutorial$^{\color{red}{\star}}$ on Reed-Solomon codes, in which a Galois Field is developed. The elements of the field are generated as consecutive powers of $X$, modulo an ...
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0
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Algorithm that solves a system of linear equations over finite fields when a parameter is needed
I was reading Kipnis' and Shamir's paper on Cryptanalysis of the HFE Public Key Cryptosystem by Relinearisation and I wanted to implement the example at the end in Octave without using any additional ...
- 1
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Finding a biggest (in terms of dimension) vector space in a finite set
I previously asked a similar question to this but noticed that the formulation was slightly different than what I am interested in, therefore I ask for any useful information on this problem (by any ...
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1
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Characteristic and order of a field [closed]
Let $\mathbb{F}$ be a finite field with characteristic $p$, prime.
Then $1$ has order $p$ in $(\mathbb{F}, +)$. I don't understand why this then implies that $p$ divides the order of $\mathbb{F}$?
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5
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Do these 25 gaussian integers make a finite field?
$$
\{ a + b i \mid a \in \{0, 1, 2, 3, 4\}, b \in \{0, 1, 2, 3, 4\} \}
$$
With calculation done in $\pmod{5}$, I'm wondering if this makes a finite field.
I thought the answer is yes at first.
Then ...
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For p prime, show ${pn \choose pj} \equiv {n \choose j}$ (mod p) for any j = 0, 1, ..., n by considering $(1 + x^p)^n$ over $\mathbb{F}_p[x]$ [duplicate]
This is a homework question I've been stuck on for the better chunk of a day, and I feel like I'm overlooking something obvious.
So far I've used the fact that in a finite field with characteristic p, ...
