Proving Ring Isomorphism of Q[x]/<x^2-2> and Q[sqrt2]

In summary, the easiest way to show that Q[x]/<x^2-2> is ring isomorphic to Q[sqrt2]={a+b(sqrt2)|a,b in Q} is to simply write down the isomorphism, which is f(a+bx)=a+b(sqrt2). This can be seen as obvious since one side only has x as a special quantity while the other has sqrt(2), and anything in Q[x]/(x^2-2) is of the form a+bx.
  • #1
math-chick_41
34
0
what is the easiest way to show that
Q[x]/<x^2-2> is ring isomorphic to
Q[sqrt2]={a+b(sqrt2)|a,b in Q}

just give me a hint how to start
 
Physics news on Phys.org
  • #2
anyone?

do I have to show that x^2 - 2 is in the kernal?
 
  • #3
why don't you just write down the (obvious) isomorphism? (obvious in the sense of one side only has x as a special quantity, the other sqrt(2), and anything in Q[x]/(x^2-2) is of the form a+bx, isn't it...)
 
Last edited:
  • #4
yes! thank you. the example in the book goes into too much detail and I was trying follow that, but yes the function f(a+bx)=a+b(sqrt2) is a ring isomorphism.
 

What is the definition of ring isomorphism?

Ring isomorphism is a mathematical concept that refers to a special type of function between two rings, which preserves their algebraic structure. In other words, if two rings are isomorphic, their elements and operations can be mapped onto each other in a way that maintains the same relationships.

Why is proving ring isomorphism important?

Proving ring isomorphism is important because it allows us to show that two seemingly different rings are actually structurally equivalent. This means that we can use properties and operations from one ring to solve problems in the other, making our mathematical work more efficient and comprehensive.

How do I prove ring isomorphism?

To prove ring isomorphism, you need to show that there exists a bijective homomorphism between the two rings. This means that the function must be both one-to-one and onto, and must preserve the operations of addition and multiplication.

What is the connection between Q[x]/ and Q[sqrt2]?

The quotient ring Q[x]/ and the field Q[sqrt2] are isomorphic, meaning that they have the same algebraic structure. This is because Q[x]/ is the ring of polynomials with rational coefficients modulo the ideal generated by x^2-2, while Q[sqrt2] is the field of numbers of the form a+b*sqrt2, where a and b are rational numbers.

How does the isomorphism between Q[x]/ and Q[sqrt2] impact mathematical computations?

The isomorphism between Q[x]/ and Q[sqrt2] allows us to use the properties and operations of one ring to solve problems in the other. This can make certain mathematical computations, such as finding roots of polynomials, more efficient and straightforward.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
148
  • Calculus and Beyond Homework Help
Replies
1
Views
974
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
503
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
948
  • Calculus and Beyond Homework Help
Replies
6
Views
765
  • Calculus and Beyond Homework Help
Replies
24
Views
673
Back
Top