Trouble with two Galois theory questions

  • Context: MHB 
  • Thread starter Thread starter LARaiders20
  • Start date Start date
  • Tags Tags
    Theory
Click For Summary
SUMMARY

The discussion focuses on two Galois theory questions involving field extensions and minimal polynomials. The first question requires finding the degree of the extension [Q(√7, √5) : Q] by determining the minimal polynomial f(x) = (x² - 7)(x² - 5). The second question involves proving that φ(4√3) = ±4√3, where φ is an element of the Galois group Gal(Q(4√3) | Q) and requires understanding the action of φ on the roots of the minimal polynomial.

PREREQUISITES
  • Understanding of Galois theory concepts, specifically field extensions
  • Familiarity with minimal polynomials and their properties
  • Knowledge of the Galois group and its actions on roots
  • Proficiency in algebraic manipulation involving square roots and polynomials
NEXT STEPS
  • Study the properties of minimal polynomials in Galois theory
  • Learn about the structure of Galois groups and their actions on field extensions
  • Explore examples of field extensions and their degrees in algebra
  • Investigate the implications of Galois theory in solving polynomial equations
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone studying Galois theory and field extensions will benefit from this discussion.

LARaiders20
Messages
2
Reaction score
0
I am so confused with these two questions. Can anyone help me out?

1) Please find [Q((√7 , √5) : Q] by finding f(x) such that Q (√7 , √5) ≅ Q[x]/(f(x)),

2) Prove that φ(4root√3) = ± 4root√3, Given that φ ∈ Gal(Q(4root√3)|Q)
 
Physics news on Phys.org
1. What you have to do is to find the minimal polynomial of \sqrt{7} and \sqrt{5}, that wil give you the degree [\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}].
Remark, note that
[\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{5},\sqrt{7}),\mathbb{Q}(\sqrt{7})][\mathbb{Q}(\sqrt{7}):\mathbb{Q}]
 
1)
Try

$$f(x) = (x^2-7)(x^2-5)$$

2)

What does

$$\phi \in Gal(\mathbb{Q}(4 \sqrt{3})|\mathbb{Q})$$ do to the roots of the minimal polynomial?
 

Similar threads

Graduate Solve Q(√7,√5) and Prove φ(4√3) = ± 4√3 | Galois Theory Help
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Generators of Galois group of ## X^n - \theta ##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Correct constructed example of Abelian Monoid?
  • · Replies 0 ·
Replies
0
Views
994
MHB Galois Groups .... A&F Example 47.7 .... ....
  • · Replies 2 ·
Replies
2
Views
1K
MHB How can we compute the Galois group of a subgroup of a splitting field?
  • · Replies 1 ·
Replies
1
Views
2K
High School Galois Groups .... A&F Example 47.7 .... ....
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Galois Theory - Structure Within Aut(K/Q) ....
  • · Replies 6 ·
Replies
6
Views
3K
MHB Fields and Field Extensions - Lovett, Chapter 7 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB What Determines the Galois Group of a Polynomial's Splitting Field?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
2K