Generators of $mathbb{Z}_2[x][x^3+x^2+1]$
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I need to find all generators of the field $mathbb{Z}_2[x][x^3+x^2+1]^*$
The star is defined as follows: $ F[x]^*m(x) = { a(x) in F[x]m(x) | gcd(a(x), m(x))=1}$
So this means we only look at the irreducible polynoms as possible generators.
The new set contains this polynoms: $1, x, x+1, x^2+x+1$.
So the set has order 4.
The number of generators is equal to the number of co-prime numbers less then 4. So there are two generators (as 1, 3 are co-prime to 4).
I know that 1 can't be a generator so I only need to check $x, x+1, x^2+x+1$.
Is this correct up to this point?
Is there a clever way to check the remaining polynomials instead of just check all the powers?
abstract-algebra finite-fields irreducible-polynomials
asked Jan 7 at 10:30
Johny DowJohny Dow
324
$endgroup$
|
show 2 more comments
$begingroup$
I need to find all generators of the field $mathbb{Z}_2[x][x^3+x^2+1]^*$
The star is defined as follows: $ F[x]^*m(x) = { a(x) in F[x]m(x) | gcd(a(x), m(x))=1}$
So this means we only look at the irreducible polynoms as possible generators.
The new set contains this polynoms: $1, x, x+1, x^2+x+1$.
So the set has order 4.
The number of generators is equal to the number of co-prime numbers less then 4. So there are two generators (as 1, 3 are co-prime to 4).
I know that 1 can't be a generator so I only need to check $x, x+1, x^2+x+1$.
Is this correct up to this point?
Is there a clever way to check the remaining polynomials instead of just check all the powers?
abstract-algebra finite-fields irreducible-polynomials
asked Jan 7 at 10:30
Johny DowJohny Dow
324
$endgroup$
$begingroup$
Possible duplicate: math.stackexchange.com/questions/3048263/…
$endgroup$
– Wuestenfux
Jan 7 at 13:09
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I can't understand your definition of $*$: you seem ot be looking at all polynomials which are multiples of $m(x)$ and are coprime to $m(x)$?
$endgroup$
– ancientmathematician
Jan 7 at 14:31
$begingroup$
I'm looking at all polynomials which are in the Field mod $x^3+x^2+1$ and co-prime to $x^3+x^2+1$.
$endgroup$
– Johny Dow
Jan 7 at 15:56
$begingroup$
The group has seven elements. In addition to the four you listed there are the cosets of $x^2+x$, $x^2+1$ and $x^2$. What do you know about groups of order seven? Hint: seven is a prime number.
$endgroup$
– Jyrki Lahtonen
Jan 7 at 19:28
1
$begingroup$
Correct. In a cyclic group of prime order any element other than the neutral element generates the whole group.
$endgroup$
– Jyrki Lahtonen
Jan 8 at 9:56
|
show 2 more comments
$begingroup$
I need to find all generators of the field $mathbb{Z}_2[x][x^3+x^2+1]^*$
The star is defined as follows: $ F[x]^*m(x) = { a(x) in F[x]m(x) | gcd(a(x), m(x))=1}$
So this means we only look at the irreducible polynoms as possible generators.
The new set contains this polynoms: $1, x, x+1, x^2+x+1$.
So the set has order 4.
The number of generators is equal to the number of co-prime numbers less then 4. So there are two generators (as 1, 3 are co-prime to 4).
I know that 1 can't be a generator so I only need to check $x, x+1, x^2+x+1$.
Is this correct up to this point?
Is there a clever way to check the remaining polynomials instead of just check all the powers?
abstract-algebra finite-fields irreducible-polynomials
asked Jan 7 at 10:30
Johny DowJohny Dow
324
$endgroup$
I need to find all generators of the field $mathbb{Z}_2[x][x^3+x^2+1]^*$
The star is defined as follows: $ F[x]^*m(x) = { a(x) in F[x]m(x) | gcd(a(x), m(x))=1}$
So this means we only look at the irreducible polynoms as possible generators.
The new set contains this polynoms: $1, x, x+1, x^2+x+1$.
So the set has order 4.
The number of generators is equal to the number of co-prime numbers less then 4. So there are two generators (as 1, 3 are co-prime to 4).
I know that 1 can't be a generator so I only need to check $x, x+1, x^2+x+1$.
Is this correct up to this point?
Is there a clever way to check the remaining polynomials instead of just check all the powers?
abstract-algebra finite-fields irreducible-polynomials
abstract-algebra finite-fields irreducible-polynomials
asked Jan 7 at 10:30
Johny DowJohny Dow
324
asked Jan 7 at 10:30
Johny DowJohny Dow
324
asked Jan 7 at 10:30
Johny DowJohny Dow
324
asked Jan 7 at 10:30
Johny DowJohny Dow
324
asked Jan 7 at 10:30
Johny DowJohny Dow
324
324
$begingroup$
Possible duplicate: math.stackexchange.com/questions/3048263/…
$endgroup$
– Wuestenfux
Jan 7 at 13:09
$begingroup$
I can't understand your definition of $*$: you seem ot be looking at all polynomials which are multiples of $m(x)$ and are coprime to $m(x)$?
$endgroup$
– ancientmathematician
Jan 7 at 14:31
$begingroup$
I'm looking at all polynomials which are in the Field mod $x^3+x^2+1$ and co-prime to $x^3+x^2+1$.
$endgroup$
– Johny Dow
Jan 7 at 15:56
$begingroup$
The group has seven elements. In addition to the four you listed there are the cosets of $x^2+x$, $x^2+1$ and $x^2$. What do you know about groups of order seven? Hint: seven is a prime number.
$endgroup$
– Jyrki Lahtonen
Jan 7 at 19:28
1
$begingroup$
Correct. In a cyclic group of prime order any element other than the neutral element generates the whole group.
$endgroup$
– Jyrki Lahtonen
Jan 8 at 9:56
|
show 2 more comments
$begingroup$
Possible duplicate: math.stackexchange.com/questions/3048263/…
$endgroup$
– Wuestenfux
Jan 7 at 13:09
$begingroup$
I can't understand your definition of $*$: you seem ot be looking at all polynomials which are multiples of $m(x)$ and are coprime to $m(x)$?
$endgroup$
– ancientmathematician
Jan 7 at 14:31
$begingroup$
I'm looking at all polynomials which are in the Field mod $x^3+x^2+1$ and co-prime to $x^3+x^2+1$.
$endgroup$
– Johny Dow
Jan 7 at 15:56
$begingroup$
The group has seven elements. In addition to the four you listed there are the cosets of $x^2+x$, $x^2+1$ and $x^2$. What do you know about groups of order seven? Hint: seven is a prime number.
$endgroup$
– Jyrki Lahtonen
Jan 7 at 19:28
1
$begingroup$
Correct. In a cyclic group of prime order any element other than the neutral element generates the whole group.
$endgroup$
– Jyrki Lahtonen
Jan 8 at 9:56
$begingroup$
Possible duplicate: math.stackexchange.com/questions/3048263/…
$endgroup$
– Wuestenfux
Jan 7 at 13:09
$begingroup$
Possible duplicate: math.stackexchange.com/questions/3048263/…
$endgroup$
– Wuestenfux
Jan 7 at 13:09
$begingroup$
I can't understand your definition of $*$: you seem ot be looking at all polynomials which are multiples of $m(x)$ and are coprime to $m(x)$?
$endgroup$
– ancientmathematician
Jan 7 at 14:31
$begingroup$
I can't understand your definition of $*$: you seem ot be looking at all polynomials which are multiples of $m(x)$ and are coprime to $m(x)$?
$endgroup$
– ancientmathematician
Jan 7 at 14:31
$begingroup$
I'm looking at all polynomials which are in the Field mod $x^3+x^2+1$ and co-prime to $x^3+x^2+1$.
$endgroup$
– Johny Dow
Jan 7 at 15:56
$begingroup$
I'm looking at all polynomials which are in the Field mod $x^3+x^2+1$ and co-prime to $x^3+x^2+1$.
$endgroup$
– Johny Dow
Jan 7 at 15:56
$begingroup$
The group has seven elements. In addition to the four you listed there are the cosets of $x^2+x$, $x^2+1$ and $x^2$. What do you know about groups of order seven? Hint: seven is a prime number.
$endgroup$
– Jyrki Lahtonen
Jan 7 at 19:28
$begingroup$
The group has seven elements. In addition to the four you listed there are the cosets of $x^2+x$, $x^2+1$ and $x^2$. What do you know about groups of order seven? Hint: seven is a prime number.
$endgroup$
– Jyrki Lahtonen
Jan 7 at 19:28
1
1
$begingroup$
Correct. In a cyclic group of prime order any element other than the neutral element generates the whole group.
$endgroup$
– Jyrki Lahtonen
Jan 8 at 9:56
$begingroup$
Correct. In a cyclic group of prime order any element other than the neutral element generates the whole group.
$endgroup$
– Jyrki Lahtonen
Jan 8 at 9:56
|
show 2 more comments
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$begingroup$
Possible duplicate: math.stackexchange.com/questions/3048263/…
$endgroup$
– Wuestenfux
Jan 7 at 13:09
$begingroup$
I can't understand your definition of $*$: you seem ot be looking at all polynomials which are multiples of $m(x)$ and are coprime to $m(x)$?
$endgroup$
– ancientmathematician
Jan 7 at 14:31
$begingroup$
I'm looking at all polynomials which are in the Field mod $x^3+x^2+1$ and co-prime to $x^3+x^2+1$.
$endgroup$
– Johny Dow
Jan 7 at 15:56
$begingroup$
The group has seven elements. In addition to the four you listed there are the cosets of $x^2+x$, $x^2+1$ and $x^2$. What do you know about groups of order seven? Hint: seven is a prime number.
$endgroup$
– Jyrki Lahtonen
Jan 7 at 19:28
1
$begingroup$
Correct. In a cyclic group of prime order any element other than the neutral element generates the whole group.
$endgroup$
– Jyrki Lahtonen
Jan 8 at 9:56