All Questions
Tagged with group-isomorphism finite-fields
25 questions
0
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65
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Decomposition of Galois group into direct product using the main theorem
I have a question concerning the main theorem of Galois theory: If $K$ is a field with finite Galois extensions $M,Z$, so that $K\subset M\subset Z$, the theorem says that $$Gal(Z/K)/Gal(Z/M)\cong Gal(...
- 372
1
vote
1
answer
77
views
Isomorphism between finite fields with polynomials $a^2 + 1$ and $b^2 + b + 2$
Find an isomorphism between finite fields with irreducible polynomials $a^2 + 1$ and $b^2 + b + 2$ respectively over $F$3.
I have tried a similar process to this answer but ended up with a very ...
- 11
3
votes
1
answer
71
views
How to interpret "multiplication" in this representation of $GF(2)$?
Most students' first encounter with the concept of "isomorphism" -- probably long before they learn the word -- comes from recognizing that the rules for adding odd and even numbers have the ...
- 24.2k
2
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0
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184
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For any $n$, does there exist a Galois extension of $\mathbb{Q}$ with Galois group elementary abelian of rank $n$?
Conjecture
My Conjecture: Suppose that $A = \{1,2,3,...,n\}$ where $n \in \mathbb{Z}^+$. Then there always exists $E/ \mathbb{Q}$ such that $\mathcal{P}(A) \cong \text{Gal}(E/\mathbb{Q})$. Moreover, ...
- 893
1
vote
0
answers
73
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Isogeny, isomorphism and point correspondence between elliptic curve and its sextic twist
Let's have an elliptic curve $E$ defined as $y^2=x^3+b$ in a finite field with prime characteristic $p$ and its sextic twist $E_6$. On both curves, let's choose a "generator" point $P$ (resp....
1
vote
0
answers
201
views
Group law of elliptic curves on finite fields
If the underlying field is $\mathbb{C}$, there is a bijective map between a given elliptic curve and $\mathbb{C} \mathbin{/} \Lambda$, where $\Lambda$ is a lattice uniquely determined by the elliptic ...
- 801
1
vote
0
answers
34
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Finding an isomorphism between finite fields of same order, but different construction.
I am trying to solve a cryptography challenge where I am given a finite field $K = \mathbb{F}_{2^{113}}$ and $K' = \mathbb{F}_{(2^{113})^{4}}$ as well as $P, Q = s\cdot P$ ($s$ is the secret I am ...
- 111
0
votes
1
answer
193
views
Isomorphisms of $GF(2)$ [closed]
The additive group of $GF(2)$ is isomorphic to $\mathbb{Z}2/\mathbb{Z}$
under addition with the "carryless" addition taken modulo 2.
An appropriate relabelling of the elements ($0 \rightarrow 1$ and ...
- 159
1
vote
1
answer
2k
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Elliptic curve group of $y^2 = x^3 + 2x + 3$ over $\mathbb{F}_5$
Let $E$ be the elliptic curve of $y^2 = x^3 + 2x + 3$ over $\mathbb{F}_5$. The points of this are
$$E(\mathbb{F}_5) = \{\infty, (1,1), (1, 4), (2, 0), (3, 1), (3, 4), (4, 0)\}.$$
I thought that this ...
- 3,960
2
votes
2
answers
712
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If $A$ is of characteristic $p$ :Then in $A[x]$ we have, $[a(x)+b(x)]^p=a(x)^p+b(x)^p$
Why, if $A$ has characteristic $p$, is the following part of a proof incorrect: Proving if $\text{char}(A)=p$ then in $A[x]$, $(x+c)^p=x^p+c^p$: $(x+c)^p=x^p+c^p$ in $A[x]$ because $(a+c)^p=a^p+c^p$ ...
- 1,255
1
vote
0
answers
90
views
transmission over isomorphism finite fields
Considering three isomorphism finite fields of size 256 A, B and C; where A and B are $GF(2^8)/GF(2)$ constructed with two distinct irreducible polynomials of degree 8, and C is a tower field $GF(((2^...
- 11
1
vote
0
answers
97
views
isomorphism between multiplicative groups of quotient rings $GF(2)[x]/p(x)$
I am interested in an isomorphism between two multiplicative groups of quotient rings $GF(2)[x]/p(x)$, specifically an algorithmic method of determining a mapping between elements.
One is the group $...
- 3,169
2
votes
1
answer
106
views
Order of the kernel of the homomorphism $\phi(x) = x^{105}$
I was given the following question:
Let $\mathbb{F}$ be a finite field with $991$ elements. Determine the number of solutions $x \in \mathbb{F}$ to the equation: $$x^{105} = 1$$
My attempt:
$\mathbb{...
- 779
-1
votes
2
answers
55
views
Are all the finite abelian groups isomorphic to the pruduct of some $\mathbb{Z}_n$s and some $\mathbb{F}_q$s?
Are all the finite abelian groups isomorphic to the pruduct of some
$(\mathbb{Z}_n,+)$s and some $(\mathbb{F}_q,+)$s?
where $\mathbb{Z}_n$ denotes the residue class ring of order $n$, $\mathbb{F}...
- 1,174
2
votes
0
answers
83
views
How to determine all the isomorphisms
I am trying to find all the isomorphisms from the representation modulo p(x), where $$p(x) = X^3 + X^2 + X + 2,$$ to the representation modulo q(Y), where $$q(y) = Y^3 + 2Y^2 + 1,$$ in $$ GF(3^3) $$
...
- 35
1
vote
0
answers
42
views
$SL_2(q)$ is isomorphic to $SU_2(q)$.
I have seen at least twice in literature that $SU_2(q) \cong SL_2(q)$. How can we actually prove that these two groups are isomorphic? The problem I see is that $SL_2(q)$ is defined over $\mathbb{F}_q$...
- 727
1
vote
1
answer
381
views
Showing a field is isomorphic to another field
I'm not sure how to show this:
Show the field $\mathbb{F}_4[X] /(X^2 +X + \alpha)$ is isomorphic to the
field $\mathbb{F}_{16}=\mathbb{F}_2[x]/(x^4 + x^3 + 1)$ by explicitly constructing an ...
0
votes
2
answers
617
views
Finding Explicit isomorphism
I dont know how quadratic non residue is used to find isomorphism.
I know X^2-a and Y^2-b is irreducible polynomial in GF(p) and they are isomorphic. But how can i find explicit isomorphism?
- 445
3
votes
2
answers
930
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Finding isomorphisms between finite fields.
I'm having trouble understanding how to find isomorphisms between finite fields. In my lecture notes it uses the following theorem:
A function $f$ is an isomorphism from $GF(z^n)$ represented ...
- 349
1
vote
1
answer
397
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Does two's complement arithmetic produce a field isomorphic to $GF(2^{n}$)?
From what I understand, we have these two isomorphisms:
$(TC, +)$ is isomorphic to the cyclic group $\mathbb{Z}/2^n\mathbb{Z}$.
$(TC, *)$ is isomorphic to the multiplicative group of polynomials.
If ...
- 123
6
votes
1
answer
899
views
Order of matrix and monic irreducible polynomial over finite field
I want to verify (and prove - in case it is true) the following proposition.
Suppose $\mathbb{F}_p$ is a finite field and $m(x)$ is a monic irreducible polynomial over $\mathbb{F}_p$ with $\mathrm{...
- 219
0
votes
1
answer
287
views
What is the isomorphism between the fields $(Z_2[x]^{<3},+_{x^3+x^2+1},\times_{x^3+x^2+1})$ and $(Z_2[x]^{<3},+_{x^3+x+1},\times_{x^3+x+1})$? [closed]
They are both Galois fields of order 8.
I'm not exactly sure what the question means - how does one determine/describe an isomorphism?
- 532
3
votes
1
answer
450
views
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 ...
- 125
3
votes
4
answers
3k
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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 (\mathbb{Z/3Z})\...
- 7,210
9
votes
2
answers
604
views
Proving that $SL_2(\mathbb{Z}_5) / \{\pm I\}\simeq A_5$
I see here that one can prove that
$$
SL_2(\mathbb{Z}_5) / \{\pm I\} \simeq A_5
$$
using the First Isomorphism Theorem.
My question is how one would do that.
I know that I need a surjective ...
- 3,389