Counterexample to Isomorphic Subfields: Help Needed

  • Thread starter Mandelbroth
  • Start date
In summary, the conversation revolves around the idea of isomorphic subfields of a field being equal. The speaker is struggling to believe this and is searching for a counterexample. They mention a few possible examples, such as the field isomorphic to ##\mathbb{Z}/(2)## in ##\mathbb{F}_4## and the vector spaces ##\mathbb{R}## and ##\mathbb{R}i##, but ultimately conclude that these are not valid counterexamples. They ask for assistance in finding a quick counterexample.
  • #1
Mandelbroth
611
24
I've got this weird thought in my head that isomorphic subfields of any field are equal. I'm having trouble believing this and I'm trying to come up with a counterexample to quell my head. But, I'm having a lot of trouble coming up with such a counterexample, and I don't know why.

Can anyone think of a quick counterexample?

So far, I've thought of the field isomorphic to ##\mathbb{Z}/(2)## generated by ##0,1\in \mathbb{F}_4##. But, when I thought about it more, no other subfield is isomorphic to ##\mathbb{Z}/(2)## in ##\mathbb{F}_4##, since ##0## and ##1## would necessarily be in the subfield. Then I thought about how ##\mathbb{R}\cong \mathbb{R}i## as vector spaces, but clearly not as rings (##\mathbb{R}i\not\ni 1##).

As usual, help is greatly appreciated.
 
Physics news on Phys.org
  • #2
Notice that the inclusion k(x2,x3,...) ⊂ k(x1,x2,x3,...) is proper, yet these fields are isomorphic.
 
  • Like
Likes 1 person

1. What is a counterexample to isomorphic subfields?

A counterexample to isomorphic subfields is a specific example that disproves the statement that two subfields are isomorphic. It shows that even though the subfields may have similar structures or properties, they are not actually isomorphic.

2. How do counterexamples to isomorphic subfields help in understanding isomorphism?

Counterexamples help in understanding isomorphism by highlighting the subtle differences between two seemingly similar subfields. They provide concrete evidence that disproves the assumption that the subfields are isomorphic, allowing for a deeper understanding of the concept.

3. Can a counterexample to isomorphic subfields be used to prove that two subfields are not isomorphic?

Yes, a counterexample can be used to prove that two subfields are not isomorphic. If a counterexample exists, it shows that the two subfields cannot be isomorphic, as there is at least one property or structure that differs between them.

4. Are counterexamples to isomorphic subfields commonly used in mathematics?

Yes, counterexamples are commonly used in mathematics to disprove statements and theories. They are especially useful in understanding isomorphism, as it is a concept that can be easily misunderstood or assumed.

5. How can I find or create a counterexample to isomorphic subfields?

To find or create a counterexample, you would need to have a thorough understanding of the properties and structures of the two subfields in question. You would then need to carefully analyze and compare the two subfields to identify any differences or inconsistencies that would disprove their isomorphism. It can also be helpful to consult with other mathematicians or references for guidance.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
963
  • Linear and Abstract Algebra
Replies
14
Views
2K
  • Linear and Abstract Algebra
Replies
8
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
3K
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Math Proof Training and Practice
3
Replies
80
Views
4K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Back
Top