What are S-integral matrices for S a finite set of primes in a number field?
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2
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I am reading the paper Geometric construction of cohomology for arithmetic
groups I by Millson and Raghunathan, and unfortunately I know almost no number theory, so I am confused by the following proposition:
Specifically, what does it mean when they say "Let $S={p_1, ldots, p_m}$ be a finite set of primes of $E$ and let $G(O_S)$ be the group of $S$ integral matrices over $E$ that preserve $Q$."
What is $O_S$ exactly? I found the definition of $O_R$ for $R$ a ring to be be the ring of roots of monic polynomials with integer coefficients. But $S$ is obviously not a ring. Is the intent her to look at the subring generated by $S$ and find the roots within this subring?
What do they mean be a "finite set of primes of $E$"? When I look up the definition of a prime element of a ring it says that an element $p$ of a ring $R$ is prime "if it is nonzero, has no multiplicative inverse and satisfies the following requirement: whenever $p$ divides the product $xy$ of two elements of $R$, it also divides at least one of $x$ or $y$." But this doesn't make sense in this context since every non-zero element of a field is invertible. Do you think that the authors mean that these are irreducible elements instead? Or is there some other meaning of prime in this context?
number-theory ring-theory commutative-algebra algebraic-number-theory
asked Nov 22 at 2:59
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add a comment |
up vote
2
down vote
favorite
I am reading the paper Geometric construction of cohomology for arithmetic
groups I by Millson and Raghunathan, and unfortunately I know almost no number theory, so I am confused by the following proposition:
Specifically, what does it mean when they say "Let $S={p_1, ldots, p_m}$ be a finite set of primes of $E$ and let $G(O_S)$ be the group of $S$ integral matrices over $E$ that preserve $Q$."
What is $O_S$ exactly? I found the definition of $O_R$ for $R$ a ring to be be the ring of roots of monic polynomials with integer coefficients. But $S$ is obviously not a ring. Is the intent her to look at the subring generated by $S$ and find the roots within this subring?
What do they mean be a "finite set of primes of $E$"? When I look up the definition of a prime element of a ring it says that an element $p$ of a ring $R$ is prime "if it is nonzero, has no multiplicative inverse and satisfies the following requirement: whenever $p$ divides the product $xy$ of two elements of $R$, it also divides at least one of $x$ or $y$." But this doesn't make sense in this context since every non-zero element of a field is invertible. Do you think that the authors mean that these are irreducible elements instead? Or is there some other meaning of prime in this context?
number-theory ring-theory commutative-algebra algebraic-number-theory
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
1
planetmath.org/RingOfSintegers
– Qiaochu Yuan
Nov 22 at 3:03
@QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well?
– ಠ_ಠ
Nov 22 at 7:28
1
I think "prime of $E$" means a prime element of the ring of integers of $E$.
– Qiaochu Yuan
Nov 22 at 20:21
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am reading the paper Geometric construction of cohomology for arithmetic
groups I by Millson and Raghunathan, and unfortunately I know almost no number theory, so I am confused by the following proposition:
Specifically, what does it mean when they say "Let $S={p_1, ldots, p_m}$ be a finite set of primes of $E$ and let $G(O_S)$ be the group of $S$ integral matrices over $E$ that preserve $Q$."
What is $O_S$ exactly? I found the definition of $O_R$ for $R$ a ring to be be the ring of roots of monic polynomials with integer coefficients. But $S$ is obviously not a ring. Is the intent her to look at the subring generated by $S$ and find the roots within this subring?
What do they mean be a "finite set of primes of $E$"? When I look up the definition of a prime element of a ring it says that an element $p$ of a ring $R$ is prime "if it is nonzero, has no multiplicative inverse and satisfies the following requirement: whenever $p$ divides the product $xy$ of two elements of $R$, it also divides at least one of $x$ or $y$." But this doesn't make sense in this context since every non-zero element of a field is invertible. Do you think that the authors mean that these are irreducible elements instead? Or is there some other meaning of prime in this context?
number-theory ring-theory commutative-algebra algebraic-number-theory
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
I am reading the paper Geometric construction of cohomology for arithmetic
groups I by Millson and Raghunathan, and unfortunately I know almost no number theory, so I am confused by the following proposition:
Specifically, what does it mean when they say "Let $S={p_1, ldots, p_m}$ be a finite set of primes of $E$ and let $G(O_S)$ be the group of $S$ integral matrices over $E$ that preserve $Q$."
What is $O_S$ exactly? I found the definition of $O_R$ for $R$ a ring to be be the ring of roots of monic polynomials with integer coefficients. But $S$ is obviously not a ring. Is the intent her to look at the subring generated by $S$ and find the roots within this subring?
What do they mean be a "finite set of primes of $E$"? When I look up the definition of a prime element of a ring it says that an element $p$ of a ring $R$ is prime "if it is nonzero, has no multiplicative inverse and satisfies the following requirement: whenever $p$ divides the product $xy$ of two elements of $R$, it also divides at least one of $x$ or $y$." But this doesn't make sense in this context since every non-zero element of a field is invertible. Do you think that the authors mean that these are irreducible elements instead? Or is there some other meaning of prime in this context?
number-theory ring-theory commutative-algebra algebraic-number-theory
number-theory ring-theory commutative-algebra algebraic-number-theory
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
edited Nov 22 at 7:02
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
asked Nov 22 at 2:59
ಠ_ಠ
5,34221242
5,34221242
1
planetmath.org/RingOfSintegers
– Qiaochu Yuan
Nov 22 at 3:03
@QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well?
– ಠ_ಠ
Nov 22 at 7:28
1
I think "prime of $E$" means a prime element of the ring of integers of $E$.
– Qiaochu Yuan
Nov 22 at 20:21
add a comment |
1
planetmath.org/RingOfSintegers
– Qiaochu Yuan
Nov 22 at 3:03
@QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well?
– ಠ_ಠ
Nov 22 at 7:28
1
I think "prime of $E$" means a prime element of the ring of integers of $E$.
– Qiaochu Yuan
Nov 22 at 20:21
1
1
planetmath.org/RingOfSintegers
– Qiaochu Yuan
Nov 22 at 3:03
planetmath.org/RingOfSintegers
– Qiaochu Yuan
Nov 22 at 3:03
@QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well?
– ಠ_ಠ
Nov 22 at 7:28
@QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well?
– ಠ_ಠ
Nov 22 at 7:28
1
1
I think "prime of $E$" means a prime element of the ring of integers of $E$.
– Qiaochu Yuan
Nov 22 at 20:21
I think "prime of $E$" means a prime element of the ring of integers of $E$.
– Qiaochu Yuan
Nov 22 at 20:21
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Here $O_S$ will be the ring of integers of $E$ with the inverses of the $p_i$
adjoined. That is,
$$O_S=O_E[p_1^{-1},ldots,p_m^{-1}]$$
where $O_E$ is the ring of algebraic integers in $E$.
(This is standard jargon in algebraic number theory.)
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Here $O_S$ will be the ring of integers of $E$ with the inverses of the $p_i$
adjoined. That is,
$$O_S=O_E[p_1^{-1},ldots,p_m^{-1}]$$
where $O_E$ is the ring of algebraic integers in $E$.
(This is standard jargon in algebraic number theory.)
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
add a comment |
up vote
1
down vote
accepted
Here $O_S$ will be the ring of integers of $E$ with the inverses of the $p_i$
adjoined. That is,
$$O_S=O_E[p_1^{-1},ldots,p_m^{-1}]$$
where $O_E$ is the ring of algebraic integers in $E$.
(This is standard jargon in algebraic number theory.)
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Here $O_S$ will be the ring of integers of $E$ with the inverses of the $p_i$
adjoined. That is,
$$O_S=O_E[p_1^{-1},ldots,p_m^{-1}]$$
where $O_E$ is the ring of algebraic integers in $E$.
(This is standard jargon in algebraic number theory.)
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
Here $O_S$ will be the ring of integers of $E$ with the inverses of the $p_i$
adjoined. That is,
$$O_S=O_E[p_1^{-1},ldots,p_m^{-1}]$$
where $O_E$ is the ring of algebraic integers in $E$.
(This is standard jargon in algebraic number theory.)
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
answered Nov 22 at 3:04
Lord Shark the Unknown
99.3k958131
99.3k958131
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
add a comment |
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
Thanks! Do you happen to know of a good reference for this sort of stuff?
– ಠ_ಠ
Nov 22 at 3:39
add a comment |
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1
planetmath.org/RingOfSintegers
– Qiaochu Yuan
Nov 22 at 3:03
@QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well?
– ಠ_ಠ
Nov 22 at 7:28
1
I think "prime of $E$" means a prime element of the ring of integers of $E$.
– Qiaochu Yuan
Nov 22 at 20:21