Ring Automorphisms that fix 1. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Automorphisms of $mathbb Q(sqrt 2)$Automorphisms of $mathbbR^n$group of automorphisms of the ring $mathbbZtimesmathbbZ$Trying to understand a proof for the automorphisms of a polynomial ringAll automorphisms of splitting fieldsDetermining automorphisms of this extensionRing automorphisms of $mathbbQ[sqrt[3]5]$Automorphism of ring and isomorphism of quotient ringsThe automorphisms of the extension $mathbbQ(sqrt[4]2)/mathbbQ$.Extension theorem for field automorphismsAre all verbal automorphisms inner power automorphisms?
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Ring Automorphisms that fix 1.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Automorphisms of $mathbb Q(sqrt 2)$Automorphisms of $mathbbR^n$group of automorphisms of the ring $mathbbZtimesmathbbZ$Trying to understand a proof for the automorphisms of a polynomial ringAll automorphisms of splitting fieldsDetermining automorphisms of this extensionRing automorphisms of $mathbbQ[sqrt[3]5]$Automorphism of ring and isomorphism of quotient ringsThe automorphisms of the extension $mathbbQ(sqrt[4]2)/mathbbQ$.Extension theorem for field automorphismsAre all verbal automorphisms inner power automorphisms?
$begingroup$
This question is a follow - up to this question about Field Automorphisms of $mathbbQ[sqrt2]$.
Since $mathbbQ[sqrt2]$ is a vector space over $mathbbQ$ with basis $1, sqrt2$, I naively understand why it is the case that automorphisms $phi$ of $mathbbQ[sqrt2]$ are determined wholly by the image of $1$ and $sqrt2$. My problem is using this fact explicitly. For example, suppose I consider the automorphism $phi$ such that $phi(1) = 1$ and $phi(sqrt2) = sqrt2$, and I want to compute the value of $phi(frac32)$. I can do the following:
$$ phi(frac32) = phi(3) phi(frac12) = [phi(1) + phi(1) + phi(1)] phi(frac12) = 3phi(frac12).$$
I am unsure how to proceed from here. I would assume that it is true that $$phi(frac11 + 1) = fracphi(1)phi(1) + phi(1) = frac12,$$ but I don't know what property of ring isomorphisms would allow me to do this.
abstract-algebra ring-theory field-theory galois-theory
$endgroup$
add a comment |
$begingroup$
This question is a follow - up to this question about Field Automorphisms of $mathbbQ[sqrt2]$.
Since $mathbbQ[sqrt2]$ is a vector space over $mathbbQ$ with basis $1, sqrt2$, I naively understand why it is the case that automorphisms $phi$ of $mathbbQ[sqrt2]$ are determined wholly by the image of $1$ and $sqrt2$. My problem is using this fact explicitly. For example, suppose I consider the automorphism $phi$ such that $phi(1) = 1$ and $phi(sqrt2) = sqrt2$, and I want to compute the value of $phi(frac32)$. I can do the following:
$$ phi(frac32) = phi(3) phi(frac12) = [phi(1) + phi(1) + phi(1)] phi(frac12) = 3phi(frac12).$$
I am unsure how to proceed from here. I would assume that it is true that $$phi(frac11 + 1) = fracphi(1)phi(1) + phi(1) = frac12,$$ but I don't know what property of ring isomorphisms would allow me to do this.
abstract-algebra ring-theory field-theory galois-theory
$endgroup$
add a comment |
$begingroup$
This question is a follow - up to this question about Field Automorphisms of $mathbbQ[sqrt2]$.
Since $mathbbQ[sqrt2]$ is a vector space over $mathbbQ$ with basis $1, sqrt2$, I naively understand why it is the case that automorphisms $phi$ of $mathbbQ[sqrt2]$ are determined wholly by the image of $1$ and $sqrt2$. My problem is using this fact explicitly. For example, suppose I consider the automorphism $phi$ such that $phi(1) = 1$ and $phi(sqrt2) = sqrt2$, and I want to compute the value of $phi(frac32)$. I can do the following:
$$ phi(frac32) = phi(3) phi(frac12) = [phi(1) + phi(1) + phi(1)] phi(frac12) = 3phi(frac12).$$
I am unsure how to proceed from here. I would assume that it is true that $$phi(frac11 + 1) = fracphi(1)phi(1) + phi(1) = frac12,$$ but I don't know what property of ring isomorphisms would allow me to do this.
abstract-algebra ring-theory field-theory galois-theory
$endgroup$
This question is a follow - up to this question about Field Automorphisms of $mathbbQ[sqrt2]$.
Since $mathbbQ[sqrt2]$ is a vector space over $mathbbQ$ with basis $1, sqrt2$, I naively understand why it is the case that automorphisms $phi$ of $mathbbQ[sqrt2]$ are determined wholly by the image of $1$ and $sqrt2$. My problem is using this fact explicitly. For example, suppose I consider the automorphism $phi$ such that $phi(1) = 1$ and $phi(sqrt2) = sqrt2$, and I want to compute the value of $phi(frac32)$. I can do the following:
$$ phi(frac32) = phi(3) phi(frac12) = [phi(1) + phi(1) + phi(1)] phi(frac12) = 3phi(frac12).$$
I am unsure how to proceed from here. I would assume that it is true that $$phi(frac11 + 1) = fracphi(1)phi(1) + phi(1) = frac12,$$ but I don't know what property of ring isomorphisms would allow me to do this.
abstract-algebra ring-theory field-theory galois-theory
abstract-algebra ring-theory field-theory galois-theory
asked 5 hours ago
Solarflare0Solarflare0
10813
10813
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2 Answers
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$begingroup$
$$
2phi(frac32) = phi(3) = 3phi(1) = 3
implies
phi(frac32) =frac32
$$
Generalizing this argument gives $phi(q) = q$ for all $q in mathbb Q$.
$endgroup$
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
add a comment |
$begingroup$
Every automorphism fixes $mathbbQ$. That is, if $K$ is any field of characteristic zero, then any automorphism of $K$ fixes the unique subfield of $K$ isomorphic to $mathbbQ$.
For the proof, we assume WLOG that $mathbbQ subseteq K$. Then:
$phi$ fixes $0$ and $1$, by definition.
$phi$ fixes all positive integers, since $phi(n) = phi(1 + 1 + cdots + 1) = n phi(1) = n$.
$phi$ fixes all negative integers, since $phi(n) + phi(-n) = phi(n-n) = 0$, so $phi(-n) = -phi(n) = -n$.
$phi$ fixes all rational numbers, since $n cdot phileft(fracmnright) = phi(m) = m$, so $phileft(fracmnright) = fracmn$.
More generally, when we consider automorphisms of a field extension $K / F$, we often restrict our attention only to automorphisms which fix the base field $F$. But when $F = mathbbQ$, since all automorphisms fix $mathbbQ$, such a restriction is unnecessary.
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
oldest
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votes
$begingroup$
$$
2phi(frac32) = phi(3) = 3phi(1) = 3
implies
phi(frac32) =frac32
$$
Generalizing this argument gives $phi(q) = q$ for all $q in mathbb Q$.
$endgroup$
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
add a comment |
$begingroup$
$$
2phi(frac32) = phi(3) = 3phi(1) = 3
implies
phi(frac32) =frac32
$$
Generalizing this argument gives $phi(q) = q$ for all $q in mathbb Q$.
$endgroup$
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
add a comment |
$begingroup$
$$
2phi(frac32) = phi(3) = 3phi(1) = 3
implies
phi(frac32) =frac32
$$
Generalizing this argument gives $phi(q) = q$ for all $q in mathbb Q$.
$endgroup$
$$
2phi(frac32) = phi(3) = 3phi(1) = 3
implies
phi(frac32) =frac32
$$
Generalizing this argument gives $phi(q) = q$ for all $q in mathbb Q$.
answered 5 hours ago
lhflhf
168k11172405
168k11172405
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
add a comment |
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
$begingroup$
In the interest of clarity's sake it might be worth noting this is the multiplicative property of ring/field homomorphisms, i.e. $phi(xy)=phi(x)phi(y)$, under the consideration $3 = 3cdot 1$.
$endgroup$
– Eevee Trainer
1 hour ago
add a comment |
$begingroup$
Every automorphism fixes $mathbbQ$. That is, if $K$ is any field of characteristic zero, then any automorphism of $K$ fixes the unique subfield of $K$ isomorphic to $mathbbQ$.
For the proof, we assume WLOG that $mathbbQ subseteq K$. Then:
$phi$ fixes $0$ and $1$, by definition.
$phi$ fixes all positive integers, since $phi(n) = phi(1 + 1 + cdots + 1) = n phi(1) = n$.
$phi$ fixes all negative integers, since $phi(n) + phi(-n) = phi(n-n) = 0$, so $phi(-n) = -phi(n) = -n$.
$phi$ fixes all rational numbers, since $n cdot phileft(fracmnright) = phi(m) = m$, so $phileft(fracmnright) = fracmn$.
More generally, when we consider automorphisms of a field extension $K / F$, we often restrict our attention only to automorphisms which fix the base field $F$. But when $F = mathbbQ$, since all automorphisms fix $mathbbQ$, such a restriction is unnecessary.
$endgroup$
add a comment |
$begingroup$
Every automorphism fixes $mathbbQ$. That is, if $K$ is any field of characteristic zero, then any automorphism of $K$ fixes the unique subfield of $K$ isomorphic to $mathbbQ$.
For the proof, we assume WLOG that $mathbbQ subseteq K$. Then:
$phi$ fixes $0$ and $1$, by definition.
$phi$ fixes all positive integers, since $phi(n) = phi(1 + 1 + cdots + 1) = n phi(1) = n$.
$phi$ fixes all negative integers, since $phi(n) + phi(-n) = phi(n-n) = 0$, so $phi(-n) = -phi(n) = -n$.
$phi$ fixes all rational numbers, since $n cdot phileft(fracmnright) = phi(m) = m$, so $phileft(fracmnright) = fracmn$.
More generally, when we consider automorphisms of a field extension $K / F$, we often restrict our attention only to automorphisms which fix the base field $F$. But when $F = mathbbQ$, since all automorphisms fix $mathbbQ$, such a restriction is unnecessary.
$endgroup$
add a comment |
$begingroup$
Every automorphism fixes $mathbbQ$. That is, if $K$ is any field of characteristic zero, then any automorphism of $K$ fixes the unique subfield of $K$ isomorphic to $mathbbQ$.
For the proof, we assume WLOG that $mathbbQ subseteq K$. Then:
$phi$ fixes $0$ and $1$, by definition.
$phi$ fixes all positive integers, since $phi(n) = phi(1 + 1 + cdots + 1) = n phi(1) = n$.
$phi$ fixes all negative integers, since $phi(n) + phi(-n) = phi(n-n) = 0$, so $phi(-n) = -phi(n) = -n$.
$phi$ fixes all rational numbers, since $n cdot phileft(fracmnright) = phi(m) = m$, so $phileft(fracmnright) = fracmn$.
More generally, when we consider automorphisms of a field extension $K / F$, we often restrict our attention only to automorphisms which fix the base field $F$. But when $F = mathbbQ$, since all automorphisms fix $mathbbQ$, such a restriction is unnecessary.
$endgroup$
Every automorphism fixes $mathbbQ$. That is, if $K$ is any field of characteristic zero, then any automorphism of $K$ fixes the unique subfield of $K$ isomorphic to $mathbbQ$.
For the proof, we assume WLOG that $mathbbQ subseteq K$. Then:
$phi$ fixes $0$ and $1$, by definition.
$phi$ fixes all positive integers, since $phi(n) = phi(1 + 1 + cdots + 1) = n phi(1) = n$.
$phi$ fixes all negative integers, since $phi(n) + phi(-n) = phi(n-n) = 0$, so $phi(-n) = -phi(n) = -n$.
$phi$ fixes all rational numbers, since $n cdot phileft(fracmnright) = phi(m) = m$, so $phileft(fracmnright) = fracmn$.
More generally, when we consider automorphisms of a field extension $K / F$, we often restrict our attention only to automorphisms which fix the base field $F$. But when $F = mathbbQ$, since all automorphisms fix $mathbbQ$, such a restriction is unnecessary.
answered 5 hours ago
60056005
37.1k752127
37.1k752127
add a comment |
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