Prove equality of number fields

  • Thread starter Thread starter shabbado
  • Start date Start date
  • Tags Tags
    Fields
Click For Summary

Homework Help Overview

The discussion revolves around proving the equality of two number fields, specifically Q[i + sqrt(2)] and Q[i][sqrt(2)], where Q represents the rational numbers.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss proving that i + sqrt(2) is in Q[i][sqrt(2)] and the reverse inclusion. There are mentions of using degree counting as a potential method, along with questions about what constitutes counting bases in this context.

Discussion Status

The discussion is ongoing with various approaches being explored. Some participants are seeking clarification on the concept of field extensions and how they relate to proving the equality of the number fields.

Contextual Notes

There are indications of confusion regarding the professor's explanations and the participants' understanding of degree counting and field extensions.

shabbado
Messages
6
Reaction score
0
Hello everyone,

I need to prove that Q[i + sqrt(2)] = Q[squrt(2)]

where Q = rationals

Any help would be appreciated.

Thanks
 
Physics news on Phys.org
Prove i+sqrt(2) is in Q[sqrt(2)]. That's easy. Then prove i and sqrt(2) are in Q[i+sqrt(2)]. That's a little harder, but not much.
 
Degree counting could work too.
 
Hurkyl said:
Degree counting could work too.

thank you for taking the time to reply.

With degree counting, would that be counting the bases? Unforunately my professor did not explain this well and I am having a difficult time finding information regarding this subject.
 
I'm not sure what you mean by "counting the bases". I'm referring to looking at the degrees of various field extensions.
 
a couple of things...

How exactly do I go about find the various field extensions and once I do, how does this help me prove equality?
 
I'm not really sure what Hurkyl is up to, but just try the direct approach. Show Q[i+sqrt(2)] is a subset of Q[sqrt(2)] and conversely.
 

Similar threads

Prove that l^p is a subset of l^q for all p,q from 1 to infinity
  • · Replies 4 ·
Replies
4
Views
3K
Proving Completeness property of ##\mathbb{R}## using Dedekind cuts
  • · Replies 20 ·
Replies
20
Views
4K
Is Galois Theory a Powerful Tool for Understanding Field Extensions?
  • · Replies 6 ·
Replies
6
Views
2K
Can Irrational Numbers Be Found Between Any Two Real Numbers?
  • · Replies 8 ·
Replies
8
Views
2K
Proving discontinuity for rational numbers (reduced form)
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
Subfields of complex numbers and the inclusion of rational#s
  • · Replies 2 ·
Replies
2
Views
3K
Proving convergence of rational sequence
  • · Replies 2 ·
Replies
2
Views
2K
Proving the existence of a real exponential function
  • · Replies 5 ·
Replies
5
Views
2K
Prove f(y) = y for every real number y
  • · Replies 1 ·
Replies
1
Views
1K