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Analogy between the fundamental theorems of arithmetic and algebra
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For the learned mathematician it may be obvious and not worth mentioning: that the fundamental theorems of arithmetic and algebra look very similar and have to do with each other, in abbreviated form:
$$ n = p_1cdot p_2 cdot dots cdot p_k$$
$$ P(z) = z_0cdot(z_1 -z)cdot (z_2 -z) cdot dots cdot (z_k -z)$$
which makes obvious that the irreducible polynoms of first degree play the same role in $mathbbC[X]$ as do the prime numbers in $mathbbZ$ (which both are unitary rings). It also gives the wording "fundamental theorem" a specific meaning: It is stated that and how some irreducible elements build the fundaments of a structure.
What I wonder is, why this analogy is not made more explicit and stressed, e.g. in introductory expositions as in Wikipedia (where the reader even is warned not to confuse the two theorems - not further mentioning the other by a single word). Might it be a superficial and maybe misleading analogy?
abstract-algebra arithmetic
asked 43 mins ago
Hans Stricker
4,65413778
add a comment |Â
up vote
6
down vote
favorite
For the learned mathematician it may be obvious and not worth mentioning: that the fundamental theorems of arithmetic and algebra look very similar and have to do with each other, in abbreviated form:
$$ n = p_1cdot p_2 cdot dots cdot p_k$$
$$ P(z) = z_0cdot(z_1 -z)cdot (z_2 -z) cdot dots cdot (z_k -z)$$
which makes obvious that the irreducible polynoms of first degree play the same role in $mathbbC[X]$ as do the prime numbers in $mathbbZ$ (which both are unitary rings). It also gives the wording "fundamental theorem" a specific meaning: It is stated that and how some irreducible elements build the fundaments of a structure.
What I wonder is, why this analogy is not made more explicit and stressed, e.g. in introductory expositions as in Wikipedia (where the reader even is warned not to confuse the two theorems - not further mentioning the other by a single word). Might it be a superficial and maybe misleading analogy?
abstract-algebra arithmetic
asked 43 mins ago
Hans Stricker
4,65413778
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
For the learned mathematician it may be obvious and not worth mentioning: that the fundamental theorems of arithmetic and algebra look very similar and have to do with each other, in abbreviated form:
$$ n = p_1cdot p_2 cdot dots cdot p_k$$
$$ P(z) = z_0cdot(z_1 -z)cdot (z_2 -z) cdot dots cdot (z_k -z)$$
which makes obvious that the irreducible polynoms of first degree play the same role in $mathbbC[X]$ as do the prime numbers in $mathbbZ$ (which both are unitary rings). It also gives the wording "fundamental theorem" a specific meaning: It is stated that and how some irreducible elements build the fundaments of a structure.
What I wonder is, why this analogy is not made more explicit and stressed, e.g. in introductory expositions as in Wikipedia (where the reader even is warned not to confuse the two theorems - not further mentioning the other by a single word). Might it be a superficial and maybe misleading analogy?
abstract-algebra arithmetic
asked 43 mins ago
Hans Stricker
4,65413778
For the learned mathematician it may be obvious and not worth mentioning: that the fundamental theorems of arithmetic and algebra look very similar and have to do with each other, in abbreviated form:
$$ n = p_1cdot p_2 cdot dots cdot p_k$$
$$ P(z) = z_0cdot(z_1 -z)cdot (z_2 -z) cdot dots cdot (z_k -z)$$
which makes obvious that the irreducible polynoms of first degree play the same role in $mathbbC[X]$ as do the prime numbers in $mathbbZ$ (which both are unitary rings). It also gives the wording "fundamental theorem" a specific meaning: It is stated that and how some irreducible elements build the fundaments of a structure.
What I wonder is, why this analogy is not made more explicit and stressed, e.g. in introductory expositions as in Wikipedia (where the reader even is warned not to confuse the two theorems - not further mentioning the other by a single word). Might it be a superficial and maybe misleading analogy?
abstract-algebra arithmetic
abstract-algebra arithmetic
asked 43 mins ago
Hans Stricker
4,65413778
asked 43 mins ago
Hans Stricker
4,65413778
edited 36 mins ago
asked 43 mins ago
Hans Stricker
4,65413778
asked 43 mins ago
Hans Stricker
4,65413778
asked 43 mins ago
Hans Stricker
4,65413778
4,65413778
add a comment |Â
add a comment |Â
2 Answers
2
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up vote
5
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This analogy can be formalise in ring theory. The sets $mathbb C[X]$ and $mathbb Z$ are both rings, and more precisely, noetherian rings.
In noetherian rings, you have a theorem that stated that every ideal can be decomposed as the intersection of finitely many primary ideals.
Which is exactly the meaning of those two theorems.
In general: if you like this kinds of analogies, study group theory, ring theory, and really any kind of algebraic structures.
answered 35 mins ago
E. Joseph
11.4k82755
add a comment |Â
up vote
3
down vote
It is a good analogy. It turns out the both $mathbb Z$ and $mathbbC[x]$ are unique factorization domains. In the case of $mathbbC[x]$, this fact, togther with the fundamental theorem of Algebra, means what you wrote: every $p(x)inmathbbC[x]$ can be written as the product of a non-zero complex number and first degree polynomials. The same thing applies to any algebraically closed field, such as the field of algebraic numbers.
answered 21 mins ago
José Carlos Santos
125k17101187
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
This analogy can be formalise in ring theory. The sets $mathbb C[X]$ and $mathbb Z$ are both rings, and more precisely, noetherian rings.
In noetherian rings, you have a theorem that stated that every ideal can be decomposed as the intersection of finitely many primary ideals.
Which is exactly the meaning of those two theorems.
In general: if you like this kinds of analogies, study group theory, ring theory, and really any kind of algebraic structures.
answered 35 mins ago
E. Joseph
11.4k82755
add a comment |Â
up vote
5
down vote
This analogy can be formalise in ring theory. The sets $mathbb C[X]$ and $mathbb Z$ are both rings, and more precisely, noetherian rings.
In noetherian rings, you have a theorem that stated that every ideal can be decomposed as the intersection of finitely many primary ideals.
Which is exactly the meaning of those two theorems.
In general: if you like this kinds of analogies, study group theory, ring theory, and really any kind of algebraic structures.
answered 35 mins ago
E. Joseph
11.4k82755
add a comment |Â
up vote
5
down vote
up vote
5
down vote
This analogy can be formalise in ring theory. The sets $mathbb C[X]$ and $mathbb Z$ are both rings, and more precisely, noetherian rings.
In noetherian rings, you have a theorem that stated that every ideal can be decomposed as the intersection of finitely many primary ideals.
Which is exactly the meaning of those two theorems.
In general: if you like this kinds of analogies, study group theory, ring theory, and really any kind of algebraic structures.
answered 35 mins ago
E. Joseph
11.4k82755
This analogy can be formalise in ring theory. The sets $mathbb C[X]$ and $mathbb Z$ are both rings, and more precisely, noetherian rings.
In noetherian rings, you have a theorem that stated that every ideal can be decomposed as the intersection of finitely many primary ideals.
Which is exactly the meaning of those two theorems.
In general: if you like this kinds of analogies, study group theory, ring theory, and really any kind of algebraic structures.
answered 35 mins ago
E. Joseph
11.4k82755
answered 35 mins ago
E. Joseph
11.4k82755
answered 35 mins ago
E. Joseph
11.4k82755
answered 35 mins ago
E. Joseph
11.4k82755
11.4k82755
add a comment |Â
add a comment |Â
up vote
3
down vote
It is a good analogy. It turns out the both $mathbb Z$ and $mathbbC[x]$ are unique factorization domains. In the case of $mathbbC[x]$, this fact, togther with the fundamental theorem of Algebra, means what you wrote: every $p(x)inmathbbC[x]$ can be written as the product of a non-zero complex number and first degree polynomials. The same thing applies to any algebraically closed field, such as the field of algebraic numbers.
answered 21 mins ago
José Carlos Santos
125k17101187
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
add a comment |Â
up vote
3
down vote
It is a good analogy. It turns out the both $mathbb Z$ and $mathbbC[x]$ are unique factorization domains. In the case of $mathbbC[x]$, this fact, togther with the fundamental theorem of Algebra, means what you wrote: every $p(x)inmathbbC[x]$ can be written as the product of a non-zero complex number and first degree polynomials. The same thing applies to any algebraically closed field, such as the field of algebraic numbers.
answered 21 mins ago
José Carlos Santos
125k17101187
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
It is a good analogy. It turns out the both $mathbb Z$ and $mathbbC[x]$ are unique factorization domains. In the case of $mathbbC[x]$, this fact, togther with the fundamental theorem of Algebra, means what you wrote: every $p(x)inmathbbC[x]$ can be written as the product of a non-zero complex number and first degree polynomials. The same thing applies to any algebraically closed field, such as the field of algebraic numbers.
answered 21 mins ago
José Carlos Santos
125k17101187
It is a good analogy. It turns out the both $mathbb Z$ and $mathbbC[x]$ are unique factorization domains. In the case of $mathbbC[x]$, this fact, togther with the fundamental theorem of Algebra, means what you wrote: every $p(x)inmathbbC[x]$ can be written as the product of a non-zero complex number and first degree polynomials. The same thing applies to any algebraically closed field, such as the field of algebraic numbers.
answered 21 mins ago
José Carlos Santos
125k17101187
answered 21 mins ago
José Carlos Santos
125k17101187
answered 21 mins ago
José Carlos Santos
125k17101187
answered 21 mins ago
José Carlos Santos
125k17101187
125k17101187
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
add a comment |Â
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
On the other hand, every polynomial ring over a field is a UFD, so the fundamental theorem of algebra appears to be a bit more specific.
– red_trumpet
13 mins ago
add a comment |Â
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