Maximal ideals of $Bbb F_2[x]$ The 2019 Stack Overflow Developer Survey Results Are InExplicit examples of infinitely many irreducible polynomials in k[x]Let $f in mathbb F_3[X]$ be reducible, degree 4 or 5 and no roots, then a monic irreducible polynomial exists of degree 2 dividing $f$Determine all maximal and prime ideals of the polynomial ring $Bbb C[x]$Find all prime and maximal ideals of ring $mathbbZ[x,y]/langle 6, (x-2)^2, y^6rangle$.Extending a principal prime of $mathbbZ[X]$ to a maximal oneWhich of the following ideals is maximal in $mathbbZ_3[x]$Maximal and Prime Ideals in a finite Ring of FunctionsCalculating the number of irreducible polynomials over a finite fieldGiven any commutative ring $R$ with unity, $R[X]$ has infinitely many maximal ideals.Question regarding algebraic closure of$ mathbbF_2$
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Maximal ideals of $Bbb F_2[x]$
The 2019 Stack Overflow Developer Survey Results Are InExplicit examples of infinitely many irreducible polynomials in k[x]Let $f in mathbb F_3[X]$ be reducible, degree 4 or 5 and no roots, then a monic irreducible polynomial exists of degree 2 dividing $f$Determine all maximal and prime ideals of the polynomial ring $Bbb C[x]$Find all prime and maximal ideals of ring $mathbbZ[x,y]/langle 6, (x-2)^2, y^6rangle$.Extending a principal prime of $mathbbZ[X]$ to a maximal oneWhich of the following ideals is maximal in $mathbbZ_3[x]$Maximal and Prime Ideals in a finite Ring of FunctionsCalculating the number of irreducible polynomials over a finite fieldGiven any commutative ring $R$ with unity, $R[X]$ has infinitely many maximal ideals.Question regarding algebraic closure of$ mathbbF_2$
$begingroup$
Prove or Disprove:
$Bbb F_2[x]$ has uncountably many maximal ideals
- For every integer $n$, every ideal of $Bbb F_2[x]$ has only finitely many elements of degree $leq n$.
The first one is false, since any maximal ideal is generated by an irreducible polynomial over $Bbb F_2$ and the number $N_n$ of monic irreducible polynomials in $Bbb F_2[x]$ of degree $n$ is $$N_n=frac1n sum_d vert n mu(d) cdot2^fracnd$$ where $mu$ is a mobius function. It is a finite number for any given $n$. Let $A_n$ be the set containing possible monic irreducible polynomials of degree $n$. For example, $A_2=x^2+x+1$. Since each $A_i$ is countable and $$textnumber of maximal ideals=cup_n A_n$$ which is also countable. Thus first bullet is false
Is this correct? Any hint for the second one?
abstract-algebra irreducible-polynomials
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
$endgroup$
add a comment |
$begingroup$
Prove or Disprove:
$Bbb F_2[x]$ has uncountably many maximal ideals
- For every integer $n$, every ideal of $Bbb F_2[x]$ has only finitely many elements of degree $leq n$.
The first one is false, since any maximal ideal is generated by an irreducible polynomial over $Bbb F_2$ and the number $N_n$ of monic irreducible polynomials in $Bbb F_2[x]$ of degree $n$ is $$N_n=frac1n sum_d vert n mu(d) cdot2^fracnd$$ where $mu$ is a mobius function. It is a finite number for any given $n$. Let $A_n$ be the set containing possible monic irreducible polynomials of degree $n$. For example, $A_2=x^2+x+1$. Since each $A_i$ is countable and $$textnumber of maximal ideals=cup_n A_n$$ which is also countable. Thus first bullet is false
Is this correct? Any hint for the second one?
abstract-algebra irreducible-polynomials
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
$endgroup$
add a comment |
$begingroup$
Prove or Disprove:
$Bbb F_2[x]$ has uncountably many maximal ideals
- For every integer $n$, every ideal of $Bbb F_2[x]$ has only finitely many elements of degree $leq n$.
The first one is false, since any maximal ideal is generated by an irreducible polynomial over $Bbb F_2$ and the number $N_n$ of monic irreducible polynomials in $Bbb F_2[x]$ of degree $n$ is $$N_n=frac1n sum_d vert n mu(d) cdot2^fracnd$$ where $mu$ is a mobius function. It is a finite number for any given $n$. Let $A_n$ be the set containing possible monic irreducible polynomials of degree $n$. For example, $A_2=x^2+x+1$. Since each $A_i$ is countable and $$textnumber of maximal ideals=cup_n A_n$$ which is also countable. Thus first bullet is false
Is this correct? Any hint for the second one?
abstract-algebra irreducible-polynomials
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
$endgroup$
Prove or Disprove:
$Bbb F_2[x]$ has uncountably many maximal ideals
- For every integer $n$, every ideal of $Bbb F_2[x]$ has only finitely many elements of degree $leq n$.
The first one is false, since any maximal ideal is generated by an irreducible polynomial over $Bbb F_2$ and the number $N_n$ of monic irreducible polynomials in $Bbb F_2[x]$ of degree $n$ is $$N_n=frac1n sum_d vert n mu(d) cdot2^fracnd$$ where $mu$ is a mobius function. It is a finite number for any given $n$. Let $A_n$ be the set containing possible monic irreducible polynomials of degree $n$. For example, $A_2=x^2+x+1$. Since each $A_i$ is countable and $$textnumber of maximal ideals=cup_n A_n$$ which is also countable. Thus first bullet is false
Is this correct? Any hint for the second one?
abstract-algebra irreducible-polynomials
abstract-algebra irreducible-polynomials
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
edited Mar 30 at 13:15
Chinnapparaj R
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
asked Mar 30 at 12:44
Chinnapparaj RChinnapparaj R
6,2062929
6,2062929
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Regarding your first question:
It seems you got the right idea, althought it's not really formally correct to write "number of maximal ideals = $bigcup_n A_n$" (the left-hand side is a number, the right-hand side is a set.. etc.).
Regarding the second questions:
What does an element of degree $leq n$ look like?
Assume the statement would be true. Then it would also be true in the case of the ideal $I = mathbbF_2[X]$. And since $J subseteq mathbbF_2[X]$ for any ideal $J$ in $mathbbF_2[X]$, the statement is true if and only if it is true for $mathbb F_2 [X]$.
Hence what you really want to think about is:
How many polynomials $p in mathbb F_2[X]$ with $deg p leq n$ do exist for fixed $n in mathbb N$?
answered Mar 30 at 14:29
lushlush
757116
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
Regarding your first question:
It seems you got the right idea, althought it's not really formally correct to write "number of maximal ideals = $bigcup_n A_n$" (the left-hand side is a number, the right-hand side is a set.. etc.).
Regarding the second questions:
What does an element of degree $leq n$ look like?
Assume the statement would be true. Then it would also be true in the case of the ideal $I = mathbbF_2[X]$. And since $J subseteq mathbbF_2[X]$ for any ideal $J$ in $mathbbF_2[X]$, the statement is true if and only if it is true for $mathbb F_2 [X]$.
Hence what you really want to think about is:
How many polynomials $p in mathbb F_2[X]$ with $deg p leq n$ do exist for fixed $n in mathbb N$?
answered Mar 30 at 14:29
lushlush
757116
$endgroup$
add a comment |
$begingroup$
Regarding your first question:
It seems you got the right idea, althought it's not really formally correct to write "number of maximal ideals = $bigcup_n A_n$" (the left-hand side is a number, the right-hand side is a set.. etc.).
Regarding the second questions:
What does an element of degree $leq n$ look like?
Assume the statement would be true. Then it would also be true in the case of the ideal $I = mathbbF_2[X]$. And since $J subseteq mathbbF_2[X]$ for any ideal $J$ in $mathbbF_2[X]$, the statement is true if and only if it is true for $mathbb F_2 [X]$.
Hence what you really want to think about is:
How many polynomials $p in mathbb F_2[X]$ with $deg p leq n$ do exist for fixed $n in mathbb N$?
answered Mar 30 at 14:29
lushlush
757116
$endgroup$
add a comment |
$begingroup$
Regarding your first question:
It seems you got the right idea, althought it's not really formally correct to write "number of maximal ideals = $bigcup_n A_n$" (the left-hand side is a number, the right-hand side is a set.. etc.).
Regarding the second questions:
What does an element of degree $leq n$ look like?
Assume the statement would be true. Then it would also be true in the case of the ideal $I = mathbbF_2[X]$. And since $J subseteq mathbbF_2[X]$ for any ideal $J$ in $mathbbF_2[X]$, the statement is true if and only if it is true for $mathbb F_2 [X]$.
Hence what you really want to think about is:
How many polynomials $p in mathbb F_2[X]$ with $deg p leq n$ do exist for fixed $n in mathbb N$?
answered Mar 30 at 14:29
lushlush
757116
$endgroup$
Regarding your first question:
It seems you got the right idea, althought it's not really formally correct to write "number of maximal ideals = $bigcup_n A_n$" (the left-hand side is a number, the right-hand side is a set.. etc.).
Regarding the second questions:
What does an element of degree $leq n$ look like?
Assume the statement would be true. Then it would also be true in the case of the ideal $I = mathbbF_2[X]$. And since $J subseteq mathbbF_2[X]$ for any ideal $J$ in $mathbbF_2[X]$, the statement is true if and only if it is true for $mathbb F_2 [X]$.
Hence what you really want to think about is:
How many polynomials $p in mathbb F_2[X]$ with $deg p leq n$ do exist for fixed $n in mathbb N$?
answered Mar 30 at 14:29
lushlush
757116
edited Mar 30 at 14:34
answered Mar 30 at 14:29
lushlush
757116
answered Mar 30 at 14:29
lushlush
757116
answered Mar 30 at 14:29
lushlush
757116
757116
add a comment |
add a comment |
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