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

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  • 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;

public class Represent_Graph_Adjacency_Matrix 
{
  private final int vertices;
  private int()() adjacency_matrix;

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

  public void makeEdge(int to, int from, int edge) 
  {
      try 
      {
          adjacency_matrix(to)(from) = edge;
      }
      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 
      {
          return adjacency_matrix(to)(from);
      }
      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);
      Represent_Graph_Adjacency_Matrix graph;
      Represent_Graph_Adjacency_Matrix graph1;
      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();

          graph = new Represent_Graph_Adjacency_Matrix(v);

          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();

          graph1 = new Represent_Graph_Adjacency_Matrix(v);

          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
Can someone help me please?

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 <k, l <n $. 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

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von neumann algebras – Order isomorphic order intervals

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 $[0,x]$?