## Are the fields $mathbb {Q}$ and $mathbb {Q[sqrt2]}$ isomorphic? [duplicate]

• Is $mathbb {Q} ( sqrt {2}) cong mathbb {Q} ( sqrt {3})$?

6 replies

This is probably a stupid question, but I can't solve it myself.

I think they are not, but I cannot formally prove it. One reason they are probably not isomorphic is that $$x ^ 2-2x-1 in mathbb {Q} (x) subset mathbb {Q} ( sqrt2) (x)$$ It has no roots in $$mathbb {Q}$$, but it has its roots in $$mathbb {Q} ( sqrt2)$$.

I am not sure if my argument is valid or not. Any suggestions / suggestions would be appreciated.

## abstract algebra – How to show a regular representation of $G$ is isomorphic to the representation of permutation by conjugation if and only if the conjugation is transitive?

Def: two representations $$(V, rho), (W, tau)$$ of $$G$$ they are isomorphic if and only if there is isomorphism $$T: V a W$$ so $$forall g en G, T circ rho_g = tau_g circ T$$.

Leave $$G$$ be a finite group, that $$rho$$ Be the regular representation. Leave $$tau$$ be the permutation representation induced by $$G$$ acts by conjugation in $$G$$. It is true that $$rho$$ Y $$tau$$ Are they isomorphic if and only if conjugation is a transitive action?

I can show that the two representations are isomorphic, then the conjugation is transitive (counting the size of the orbits), but not vice versa. I could not build such $$T$$ To make them isomorphic.

## Java: how to write a program to verify if two graphics are isomorphic or not. My code accepts the entry of two graphics.

package net.codejava;

//This is a java program to represent graph as a adjacency matrix
import java.util.Scanner;

{
private final int vertices;

{
vertices = v;
adjacency_matrix = new int(vertices + 1)(vertices + 1);
}

public void makeEdge(int to, int from, int edge)
{
try
{
}
catch (ArrayIndexOutOfBoundsException index)
{
System.out.println("The vertices does not exists");
System.out.println("Both the graphs are not isomorphic");
}
}

public int getEdge(int to, int from)
{
try
{
}
catch (ArrayIndexOutOfBoundsException index)
{
System.out.println("The vertices does not exists");
System.out.println("Both the graphs are not isomorphic");
}
return -1;
}

public static void main(String args())
{
int v, e, count = 1, to = 0, from = 0;
Scanner sc = new Scanner(System.in);
try
{
System.out.println("Enter the number of vertices of Graph 1: ");
v = sc.nextInt();
System.out.println("Enter the number of edges of Graph 1: ");
e = sc.nextInt();

System.out.println("Enter the edges:  ");
while (count <= e)
{
to = sc.nextInt();
from = sc.nextInt();

graph.makeEdge(to, from, 1);
count++;
}

System.out.println("The adjacency matrix for the given graph is: ");

for (int i = 1; i <= v; i++)
{
for (int j = 1; j <= v; j++)
System.out.print(graph.getEdge(i, j) + " ");
System.out.println();
}
count=1;

System.out.println("Enter the number of vertices of Graph 2: ");
v = sc.nextInt();
System.out.println("Enter the number of edges of Graph 2: ");
e = sc.nextInt();

System.out.println("Enter the edges:  ");
while (count <= e)
{
to = sc.nextInt();
from = sc.nextInt();

graph1.makeEdge(to, from, 1);
count++;
}

System.out.println("The adjacency matrix for the given graph is: ");

for (int i = 1; i <= v; i++)
{
for (int j = 1; j <= v; j++)
System.out.print(graph1.getEdge(i, j) + " ");
System.out.println();
}

}
catch (Exception E)
{
System.out.println("Somthing went wrong");
}

sc.close();
}
}


How do I verify whether the two graphs are isomorphic or not?
I need it for my school homework
And I just want to know the way and I can code it

## The solution space of $A_ {12} X = 0$ and $B_ {12} X = 0$ is isomorphic?

Leave $$A$$ be a $$n times n$$ invertible matrix Assume
$$A = begin {pmatrix} A_ {11} & A_ {12} \ A_ {21} and A_ {22} end {pmatrix}$$
$$A ^ {- 1} = begin {pmatrix} B_ {11} and B_ {12} \ B_ {21} and B_ {22} end {pmatrix}.$$
where $$A_ {11}$$ is a $$l times k$$ matrix, $$B_ {11}$$ is a $$k times l$$ matrix, $$1 . How to show
$$W = { alpha; A_ {12} alpha = 0 }$$
Y $$U = { beta; B_ {12} beta = 0 }$$ It has the dimension, so it is isomorphic.

Intuitively, we have to show
$$(n-k) -rank A_ {12} = (n-l) -rank (B_ {12}).$$
But it sounds impossible at first sight.

## ag.algebraic geometry: can a simple vector package isomorphic to its torsion?

Leave $$V$$ be a vector pack on an algebraic curve $$C$$. It is possible that $$V cong V otimes L$$ for a line pack $$L$$? This is definitely possible if $$V$$ It is decomposable, for example if $$V cong mathcal {O} _C oplus L$$ with $$L ^ 2 cong mathcal {O} _C$$. I want to show that $$V cong V otimes L$$ it implies $$L cong mathcal {O} _C$$ Yes $$V$$ it is simple.

## abstract algebra: the finite Boolean ring with $1 neq 0$ is isomorphic to $mathbb {Z} _2 times mathbb {Z} _2 times dots times mathbb {Z} _2$

Leave $$R$$ be a finite boolean ring with $$1 neq 0$$. Show that $$R cong mathbb {Z} _2 times mathbb {Z} _2 times dots times mathbb {Z} _2$$.

This is exercise 2 on p267 in the abstract algebra book of Dummit and Foote. The author gives the hint of using that if $$e$$ is an idempotent of a ring $$R$$ with $$1$$, so

$$R cong Re times R (1-e)$$

where $$Re$$ it's a ring with identity $$e$$ Y $$R (1-e)$$ a ring with identity $$1-e$$.

My attempt:

Yes $$R = {0,1 }$$, so $$R cong mathbb {Z} _2$$ it's trivial

Therefore, we can assume that $${0,1 }$$ is properly contained in $$R$$. Therefore, we can choose an item $$x notin {0,1 }$$. We obtain because each element in a Boolean ring is an idempotent:

$$R cong Rx times R (1-x)$$

$$Rx$$ it's a ring with identity $$x neq 0$$ Y $$R (1-x)$$ it's a ring with identity $$1-x neq 0$$. Further, $$Rx$$ Y $$R (1-e)$$ have cardinality strictly smaller than the cardinality of $$R$$. These rings are also Booleans. Induction produces the result. $$quad square$$

Is this correct?

## Is $( mathbb {R}, +)$ isomorphic to a subgroup of $S_ omega$?

Is $$( mathbb {R}, +)$$ isomorphic to a subgroup of $$S_ omega$$, the group of permutations of the set of nonnegative integers. $$omega$$?

## abstract algebra – prove that two subgroups are isomorphic

Assume that $$G$$ it's a group, and that $$N$$ is a maximum normal subgroup of $$G$$. This means that if $$H$$ it's a subgroup of $$G$$ such that $$N subsetneq {H}$$, so $$H = G.$$ Assume that $$H_1$$ Y $$H_2$$ are two non-trivial subgroups of $$G$$ (that is to say. $$H_1 neq { {1 }} neq {H_2}$$) such that $${H_1} cap {N} = {1 } = {H_2} cap {N}.$$ So, it must be the case that $$H_1$$ is isomorphic to $$H_2$$.

This is an exercise in a graduate level algebra textbook. I can not think of a way to address this problem!

## Combinatorial: co-spectral fractional isomorphic graphs with different Laplacian spectra

Thank you for contributing an answer to MathOverflow!

But avoid

Leave $$M$$ Be a von Neumann algebra. Yes $$x$$ is positive, then Lemma 2.1 (3) of the document "Order Isomorphisms of Operator Intervals in von Neumann Algebras" (Mori, Integral Equations and Operator Theory, 2019, https://arxiv.org/abs/1811.01647) indicates that yes $$p$$ is the support projection of $$x$$, then the order interval $$[0,p]$$ is the isomorphic order to $$[0,x]$$ through the map $$y mapsto x ^ {1/2} and x ^ {1/2}$$. It is stated that the test here is "easy to see".
But in the previous document "Order isomorphisms of operator intervals" (Semrl, Integral equations and Operator theory, 2017, https://www.fmf.uni-lj.si/~semrl/preprints/orderoperatorintervals.pdf), Semrl demonstrates this in the special case where $$M = B (H)$$ Y $$x$$ is injective, that is, its support projection is $$I$$. Your test takes 1.5 pages (pages 38-39). I do not understand why Mori claims that his more general result is "easy to see". I need this in one of my documents, so I tried it myself, based on characterizing the inverse as the map $$y mapsto lim_n f_n (x) and f_n (x)$$ (weak * -limit) where $$f_n (t): = t ^ {- 1/2}$$ for $$t geq 1 / n$$ and 0 in other places. But this test is not trivial, one has to prove, among other things, that this limit always exists if $$0 leq and leq x$$.
I'm missing a simple argument that shows that $$y mapsto x ^ {1/2} and x ^ {1/2}$$ is an isomorphism of order between $$[0,p]$$ Y $$[0,x]$$?