Finite fields are structures arising in abstract algebra. 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 $\operatorname{GF}(q)$.

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Finite Field Isomorphism

Suppose $E$ and $E'$ are two degree-$n$ extensions of $\mathbb{F}_p$. They are both splitting fields of $x^{p^n}-x$ and are isomorphic. Is it possible to obtain an isomorphism $E\to E'$ that fixes the ...
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Computing Galois group of polynomial over finite field

What is the Galois group of $(x^2-1)(x^2-2)...(x^2-p+1)$ over $\mathbb Z_p$ for an odd prime, $p$? There are exactly $\frac{p-1}{2}$ squares in $\mathbb Z_p$ but my guess is the group is $\mathbb ...
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Reducing a matrix over a field of 3 elements

I have the augmented matrix $$\begin{bmatrix} 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ \end{bmatrix}$$ And I need to reduce ...
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Is the Characteristic polynomial of $ A $ is a Primitive polynomial?

Suppose $ A_{n\times n} $ matrix on $ \Bbb F_{p} $ ($P$ is prime) and if the Characteristic polynomial is irreducible then we can make a finite field that the order of $P^n$ with $ A_{n\times n} $ . ...
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Computational Complexity of exponentiating

I'm currently studying this paper and I am trying to understand the complexity of the interpolate algorithm, which is supposed to be $O((l+m)^2)$. So first the algorithm runs in $r$ steps where ...
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Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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Construct $K=\{ (a+bi)+(c+di)\sqrt{2} \}$ with Extension Fields

Construct $K=\{ (a+bi)+(c+di)\sqrt{2} \}$ with Extension Fields. where $i=\sqrt{-1}$. Can take it for granted that K is a field where degree is 4. construct K and confirm its degree we need ...
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How to find the minimum polynomial over a Galois Field

How would I find the minimum polynomial $m_4(x)$ of $\xi^4$, where $\xi= x \pmod{x^6+x+1}$ in $GF(2^6)$? This is what I think I need to do so far: Let $\alpha=\xi^4$. Then I would find the conjugacy ...
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What does it mean to represent elements of an ideal?

Say I have the polynomial $x^9 + 1$ Then: $x^9 + 1 = (x+1)(x^2 + x + 1)(x^6 + x^3 + 1)$ is a complete factorization over $GF(2)$ of $x^9 + 1$ The dimension of each ideal is: length $n - ...
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In what way is a quadratic extention to a finite field isomorphic to a finite field of higher order?

I have read (I don't remember where) that a finite field that is quadratically extended, say $\mathbb F_p[\sqrt 3]$ for example, is isomorphic to the finite field $\mathbb F_{p^2}$ (assuming the ...
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Separable and inseparable extensions.

I'm very confused with separable extensions. I need to prove: Let $E/F$ finite extension. Suppose there is an element $\alpha \in E$ which is not separable over $F$. Prove the existence of an ...
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32 views

Construct generator matrix given generator polynomial?

How would I take a generator polynomial and construct a generator matrix out of it for a cyclic code? For example, I have a cyclic code in: $R_{15}=GF(2)[x] / \langle x^{15} + 1\rangle$ This is ...
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Finding all elements in GF(2^4) in terms of given polynomial

I'm working with polynomials over finite fields at the moment and a question. I found this table http://www.csee.umbc.edu/~lomonaco/f97/442/Peterson_Table.html and I picked a polynomial of degree 4 ...
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29 views

Finding the basis vectors of a null space of a matrix in GF2

In this document, page 11, the author writes that the basis vectors of the null space of the following matrix \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & ...
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Galois subfields

S R Wicker in his book states ... $GF(p^m)$ has all sub-fields of order $p^b$ provided $b$ divides $m$. Qn: In order to have a sub-filed of order $p^b$ shouldn't we have also the constraint: $p^b ...
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How to construct isogenies between elliptic curves over finite fields for some simple cases?

From the theorem of Tate, it is known that the two elliptic curves over the same field $\mathbb{F}_p$ are isogenous iff they have the same number of points. For $p\equiv 3\mod 4$, the curve ...
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Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...
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Is there another methods for counting points on the curve $x^3 + y^3 =1$ over finite fields?

For the circle $(C): x^2 + y^2=1$ over finite field, we can use simple method to count the number of points. The case $p\equiv 1\mod 4$ is not difficult to find, because $-1$ is a square on $F_p$. ...
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32 views

Maximal proper subfields and the behavior of the norm

This is a more general version of my previous question (Yet another question on finite fields), which has yet to attract any comments (nevermind answers). It occurred to me that I might be getting ...
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What do $l+p$ and $lp$, where $p$ is a point and $l$ is a line, mean in geometery?

I am looking at a graph theory problem that describes the partite sets of a bipartite as two copies of the $(m+1)$-dimensional vector space over the finite field $\mathbb{F}_{p^n}$ ($p$ is prime and ...
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77 views

Is there any algorithm that produces irreducible polynomials over $\mathbb{F}_2$?

Is there any (fast) algorithm that produces irreducible polynomials over $\mathbb{F}_2$? EDIT: I look up for a irreducible polynomial generator, that is different from decider algorithm for ...
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101 views

irreducibility of a polynomial over $\mathbb{F}_2$

Is there any criterion (theorem or algorithm) to find out a given polynomial $f(x)$ is irreducible over the field $\mathbb{F}_2$?
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Yet another question on finite fields

Let $p$ be prime, and let $m,n \in \mathbb{Z}$ be such that $m \mid pn$ but $m \nmid n$. Let $N^{pn}_n$ denote the norm function mapping $\mathbb{F}_{q^{pn}} \rightarrow \mathbb{F}_{q^n}$, defined by ...
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Tensorproduct of finite fields

I perfectly understand the tensor product of vector spaces over finite fields. But when I regard these vector spaces as finite fields I get confused. Let the vector spaces $\mathbb{F}_p^m$ and ...
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Finite fields: factorization of the trace function over the base field

Let $q$ be a prime power, and $m$ a positive integer. The trace function from $GF(q^m)$ to $GF(q)$ is defined to be the mapping $$Tr : GF(q^m) \rightarrow GF(q) $$ $$Tr(x) = ...
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If $f$ and $g$ are independent polynomials and $h$ is a nonzero polynomial, then $fh$ and $gh$ are independent

I am doing some problems outside of class and have a couple of questions that I cannot figure out how to start. If $f$ and $g$ are independent polynomials and $h$ is a nonzero polynomial over $F$, ...
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Additive character sum over additive subgroups of finite fields, with special monomial arguments

Let $F$ be a finite field with $q = p^n$ elements, let $\psi$ be a non-trivial additive character of $F$, let $m$ be an integer coprime to $q-1$, and let $K$ be a large subspace of $F$, say $K$ is the ...
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Why is the image of a $\pmod p$ Galois representation finite?

Let $\overline{\mathbb{F}_q}$ be the algebraic closure of the finite field on $q=p^r$ elements, and $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ the absolute Galois group with the profinite ...
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$E/k$ is not seperable but still $E$ can be genreated by one element over $k$

Let $E$ be a finite extension of $k$ and let $p^r = [E:k]_{i}$. We assume that the characteristic is $p > 0$. Assume that there is no exponent $p^s$ with $s <r$ such that $E^{p^s}k$ is separable ...
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Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
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Irreducibility of $x^3 − a$ over the field of $q$ elements

I have done much research on this specific question. I have come across many different theorems and definitions on this topic. However, I am having difficulties piecing them together to create a nice ...
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Linear equations modulo different prime through CRT

Given prime $p$ and equations $$a_1x+b_1y+c_1z=d_1\bmod p \\ a_2x+b_2y+c_2z=d_2\bmod p \\ a_3x+b_3y+c_3z=d_3\bmod p$$ where $a_i,b_i,c_i,d_i\in\Bbb Z_p$, we can solve for $x,y,z\in\Bbb Z_p$. Now ...
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prove Y + Z := (Y ∪ Z) \ (Y ∩ Z) (Y, Z ⊆ X)???

let X be a set and P(X) be power set of X. Show that P(X)together with addition Y + Z := (Y ∪ Z) \ (Y ∩ Z) (Y, Z ⊆ X) and the scalar: 0 · Y := ∅, 1 · Y := Y (Y ⊆ X) becomes a vector space over ...
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Find root of polynomial over finite field

Let $\mathbb{k} = \mathbb{F}_2[\alpha]$, where $\alpha$ is a root of $x^4+x+1$. I'm stuck with finding roots of $x^2 + x + 1$ in $\mathbb{k}$. I'd be greateful for any advice.
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Geometric interpretation of subgroups of an elliptic curve group?

I am particularly interested in elliptic curves over finite fields of prime order, so let $\mathbb{F}_{p}$ denote the finite field of order $p$ (where $p$ is prime) and let $E$ be the elliptic curve ...
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Isomorphism to a finite field

I'm trying to show a ring S, set of 2x2 matrices in mod5 is isomorphic to the field GF(25). I was thinking for the mapping to use the Frobenius map, as it's clearly a homomorphism. If I can show it's ...
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A more general Kloosterman-type sum

Let $\mathbb{F}_q$ be a finite field and let $a,b \in \mathbb{F}_q$ not both zero. Let $\psi$ be the canonical additive character on $\mathbb{F}_q$. The classical Kloosterman sum is given by $$ K(a,b) ...
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What random process generates a uniformly random plane in $\mathbb{F}_p^m$?

Let $k = \mathbb{F}_p$ be the finite field of order $p$, and let $V = k^m$ be a finite-dimensional vector space over $k$. I would like to pick a random vector subspace of dimension 2. What random ...
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Let $L=F_2(x,y)$ and let $K=F_2(x^2,y^2)$. Show that $L/K$ is not simple. [duplicate]

Let $L=F_2(x,y)$ (the field of rational functions in two variables $x$ and $y$ over $F_2$)and let $K=F_2(x^2,y^2)$ (the subfield of L consisting of rational functions in $x^2$ and $y^2$). Show that ...
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How do finite fields work?

I know that a field must satisfy a set of axioms, the one that is causing me most discomfort is closure under addition. All the roots of $X^q-X$ are all the elements of a finite field of order $q$. ...
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Proving irreducibility over F_11, writing multi and div rules. [closed]

Could someone provide a solution to this? a. Prove that $f(x)=x^2 - 2$ is irreducible over $\mathbb{F}_{11}$. b. Write down division and multiplication rules in ...
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Calculating the generator of a Finite Field

I'm trying to understand how to compute the generator of a finite field. $\alpha \in GF(q)$ so for example if I was working with $GF(2^8)$ I think I would need to find $\alpha ^ {256-1} \equiv 1 ...
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Are $\mathbb{F}_{p^{d-1}} \subset \mathbb{F}_{p^d}$ Galois Extensions?

I know that $\mathbb{F}_{p}\subset \mathbb{F}_{p^d}$ is Galois extensions and the degree of extension $[\mathbb{F}_{p^d} : \mathbb{F}_{p}]=d$. but my question is :- $\mathbb{F}_{p^{d-1}} \subset ...
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If $\phi$ : $F_2[X]/(X^3+X+1)$ $\rightarrow$ $F_2[X]/(X^3+X^2+1)$ is an isomorphism, then prove that $\phi(a)=b+1$, with $a$, $b$ the classes of $X$ [duplicate]

Consider two fields: $K: $ $F_2[X]/(X^3+X+1)$. Let $a$ be the class of $X$ (so $a=X+(X^3+X+1))$ $L: $ $F_2[X]/(X^3+X^2+1)$. Let $b$ be the class of $X$ (so $b=X+(X^3+X^2+1))$ $F_2$ denotes the ...
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Show that $a^3+a+1$ is in the subfield $L$ with $4$ elements of the field $K=F_2[X]/(X^4+X^3+1)$

The field $K$ is constructed in the following way: $K=F_2[X]/(X^4+X^3+1)$, where $F_2$ is short for $\mathbb{Z}$/$2$$\mathbb{Z}$. Let $a$ be the class of $X$ in $K$ (so $a=X+(X^4+X^3+1))$. The ...
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Confusion between vector space, field and sets

From my understanding, a vector space is a set that is closed under addition and multiplication, so let $A$ denote a set, a vector space $V = (A, +, \times)$ But whenever you read the definition of ...
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105 views

Let $(F,+,\cdot)$ is the finite field with $9$ elements. Then which of the following are true? [closed]

Let $(F,+,\cdot)$ is the finite field with $9$ elements. Let $G=(F,+)$ and $H=(F-\{0\}, \cdot)$ denote the underlying additive and multiplicative groups respectively. Then, $G\cong ...
2
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2answers
46 views

find out what $[\mathbb{Q}(a^2 + 2)/ \mathbb{Q} ] $ when $a$ satisfies $\alpha^6 - 3 \alpha^3 - 6 = 0$

find out what $[\mathbb{Q}(a^2 + 2)/ \mathbb{Q} ] $ when $a$ satisfies $\alpha^6 - 3 \alpha^3 - 6 = 0$ using the tower law. I have figured out that $[\mathbb{Q}(a)/ \mathbb{Q} ] = 6$, but I am unsure ...
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29 views

Reduced polynomial has the same Galois group as the original

I'm trying to prove that for any monic $f(x)=a_0+\dots +a_{n-1}x^{n-1}+x^n\in F[x]$, we have that the reduced polynomial $$g(x)=f(\frac{x-a_{n-1}}{n})$$ has the same Galois group as the original ...
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gcd of two polynomials over the gf(2^4)

Can someone show me step by step of "Extended Euclidean Algorithm" Multiplicative inverse of polynomial gcd of $(x^4+x+1)$ and $(x^2+1)$ over $gf(2^4)$? what I did : $(x^4+x+1) = ...