Q(sqrt(2)) and Q(sqrt(3)) not isomorphic?

  • Context: Graduate 
  • Thread starter Thread starter geor
  • Start date Start date
Click For Summary
SUMMARY

The discussion confirms that the fields Q(√2) and Q(√3) are not isomorphic, despite both being vector spaces over Q with dimension 2. The key distinction lies in the fact that while they can be isomorphic as vector spaces, they do not maintain field structure, specifically regarding the product operation. The proof involves demonstrating that any isomorphism must fix Q and that √2 cannot be mapped to a rational number, leading to the conclusion that no field isomorphism exists between Q(√2) and Q(√3). This principle extends to any square-free integers m and n.

PREREQUISITES
  • Understanding of field theory and vector spaces
  • Familiarity with Galois theory concepts
  • Knowledge of isomorphisms in algebra
  • Basic properties of square roots and rational numbers
NEXT STEPS
  • Study the properties of field extensions in Galois theory
  • Learn about the structure of vector spaces over fields
  • Explore examples of non-isomorphic fields with different square-free integers
  • Investigate the concept of field isomorphisms and their implications in algebra
USEFUL FOR

Students of algebra, particularly those studying Galois theory, mathematicians interested in field theory, and educators looking for examples of non-isomorphic fields.

geor
Messages
35
Reaction score
0
Hello all,

I am studying Algebra and in the chapter where Galois theory is introduced, I
see the following exercise:

"Prove that Q(sqrt(2)) and Q(sqrt(3)) are not isomorphic"

Well, It seems that I am a bit behind because I really don't get it... :(
I mean, I'm sure that this is the case, since it is a question in the book
(and surely 'not' is not a typo!), but these are vector spaces over Q,
both of dimension 2, so shouldn't they be isomorphic by sending
sqrt(2) to sqrt(3) and any rational number to itself?!

What do I miss here?

Thanks a lot in advance..
 
Last edited:
Physics news on Phys.org
Ooops! I think I see it now..
They are isomorphic as vector spaces but not as fields, right?
The isomorphism I said above does not respect the product..

That's it, right?!
 
Right. (Have you yet shown there isn't a field isomorphism?)
 
What more is true, is that given any square free intgers m and n, Q(sqrt(m)) and Q(sqrt(n)) are nonisomorphic. Intution serves right when you say that it "does not respect the product" but being more rigorous, show that no ismorphism can possibly exist between the two fields by first showing that any isomorphism fixes Q and that sqrt 2 (in this specific case) cannot be sent to any rational number, ie. sqrt 2 is sent to a+b*sqrt 3 for some nonzero rational b. This proof easily generalizes to square free m and n.
 
Thanks a lot!
 

Similar threads

Undergrad Show ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
5K
Undergrad Determining isomorphism for ##\frac{R}{(a, b)}##
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
  • · Replies 31 ·
2
Replies
31
Views
2K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
1K
Undergrad Meaning of the notations: ##\mathbb{Z}[\frac{1}{a}]##
  • · Replies 1 ·
Replies
1
Views
994
Undergrad Galois Theory - Structure Within Aut(K/Q) ....
  • · Replies 6 ·
Replies
6
Views
3K
MHB Galois Groups .... A&F Example 47.7 .... ....
  • · Replies 2 ·
Replies
2
Views
1K
High School Galois Groups .... A&F Example 47.7 .... ....
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Questions about non existence of GCDs vs (coimages, cokernels)
  • · Replies 12 ·
Replies
12
Views
755