---
# System prepended metadata

title: 'Galois Theory [ch.9]'
tags: [math, abstract algebra]

---


# Galois Theory [ch.9]

## 9.1. What Is Galois Theory?
- **Definition 9.1: Algebraic Extension**
![Screenshot 2024-11-30 at 9.23.35 PM](https://hackmd.io/_uploads/BkpZ4tOQJg.png)
Note: This means $\alpha \in K$ is a root of some polynomials in $F$

## 9.2. A Quick Review of Polynomials and Field Extensions
- check [chapter 5](https://hackmd.io/7rwr4TX5RIepD_D0JsuGnA) and [chapter 8](https://hackmd.io/uJXu_vLfSWKV42LuNi3n3Q).

## 9.3. Fields of Algebraic Numbers
- **Theorem 9.3**
![Screenshot 2024-11-30 at 9.28.49 PM](https://hackmd.io/_uploads/Bk-QBKdXyg.png)
![Screenshot 2024-11-30 at 9.34.06 PM](https://hackmd.io/_uploads/BkILLYOXJg.png)
Note: 
    - $F[\alpha]$ is the smallest subring of $K$ contains $\alpha$ and $F$.
    - $F(\alpha)$ is the smallest subfield of $K$ contains $\alpha$ and $F$
- **Corollary 9.5**
![Screenshot 2024-11-30 at 9.34.41 PM](https://hackmd.io/_uploads/ByMdIKuXke.png)
Note$^2$: The converse of Corollary 9.5 is false; see Exercise 9.7. 
- **Theorem 9.6**
![Screenshot 2024-11-30 at 9.35.11 PM](https://hackmd.io/_uploads/HymaIKO7kl.png)
- **Definition 9.8: Minimal Polynomial of $\alpha$ over $F$**
![Screenshot 2024-11-30 at 9.43.45 PM](https://hackmd.io/_uploads/Hymqutd7Je.png)
![Screenshot 2024-11-30 at 9.44.07 PM](https://hackmd.io/_uploads/H1oyYt_7yg.png)
Note$^4$: We are using the fact that every ideal in F [x] is principal; see Theorem 5.21.
- **Theorem 9.10**
![Screenshot 2024-11-30 at 9.45.41 PM](https://hackmd.io/_uploads/B1qWKt_71g.png)

## 9.4. Algebraically Closed Fields
- **Definition 9.11: Algebraically Closed**
![Screenshot 2024-11-30 at 9.46.47 PM](https://hackmd.io/_uploads/Bke8YKOm1x.png)
Note: For example, the field $\mathbb{C}$ is algebraically closed.
- **Proposition 9.12**
![Screenshot 2024-11-30 at 9.48.29 PM](https://hackmd.io/_uploads/Hk6jtKdXyg.png)
- **Definition 9.14: Algebraic Closures**
![Screenshot 2024-11-30 at 9.51.10 PM](https://hackmd.io/_uploads/HJ1L9F_QJx.png)
- **Theorem 9.15**
![Screenshot 2024-11-30 at 9.51.43 PM](https://hackmd.io/_uploads/ByWOctd71e.png)
![Screenshot 2024-11-30 at 9.52.42 PM](https://hackmd.io/_uploads/HJso5F_7Jl.png)

## 9.5. Automorphisms of Fields
- **Example 9.18**
![Screenshot 2024-11-30 at 10.34.16 PM](https://hackmd.io/_uploads/SyS_EcOm1g.png)
- **Definition 9.19: Galois Groups**
![Screenshot 2024-11-30 at 10.33.21 PM](https://hackmd.io/_uploads/B1gSNc_71x.png)
![Screenshot 2024-11-30 at 10.37.19 PM](https://hackmd.io/_uploads/HJyXB5_Qyl.png)
- **Proposition 9.20**
![Screenshot 2024-11-30 at 10.34.46 PM](https://hackmd.io/_uploads/rk6K45d7Jx.png)
![Screenshot 2024-11-30 at 10.36.10 PM](https://hackmd.io/_uploads/rkjyB5_7kx.png)
- **Proposition 9.22**
![Screenshot 2024-11-30 at 10.36.55 PM](https://hackmd.io/_uploads/B1RXScdm1g.png)

## 9.6. Splitting Fields — Part 1
- **Lemma 9.23**
![Screenshot 2024-11-30 at 10.38.58 PM](https://hackmd.io/_uploads/BJbFBqOm1x.png)
![Screenshot 2024-11-30 at 10.39.24 PM](https://hackmd.io/_uploads/BJaqHcu7Jg.png)
- **Theorem 9.25**
![Screenshot 2024-11-30 at 10.42.02 PM](https://hackmd.io/_uploads/SyiN8qOXkx.png)
- **Corollary 9.26**
![Screenshot 2024-11-30 at 10.43.12 PM](https://hackmd.io/_uploads/S1-q8cu7Jx.png)
- **Corollary 9.27**
![Screenshot 2024-11-30 at 10.43.50 PM](https://hackmd.io/_uploads/rJjsLquQye.png)
Note: 
    - check textbook to see an example - **9.6.1. The splitting field of $X^4 − 2$ and its automorphisms.**
    - For the example above, we have to know $[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]=4$, since we need the basis $\{1, 2^{1/4} ,2^{2/4}, 2^{3/4}\}$ with scalars in $\mathbb{Q}$ to form $\mathbb{Q}(\sqrt[4]{2})$

## 9.7. Splitting Fields — Part 2
- **Theorem 9.28: Galois Extensions**
![Screenshot 2024-11-30 at 11.17.22 PM](https://hackmd.io/_uploads/B1mt0cOXyx.png)
- **Lemma 9.31**
![Screenshot 2024-11-30 at 11.17.54 PM](https://hackmd.io/_uploads/r1TjRcuXke.png)
![Screenshot 2024-11-30 at 11.21.31 PM](https://hackmd.io/_uploads/BJlYkodQyg.png)
- **Proposition 9.34**
![Screenshot 2024-11-30 at 11.22.51 PM](https://hackmd.io/_uploads/S1MAJouQJx.png)
![Screenshot 2024-11-30 at 11.23.17 PM](https://hackmd.io/_uploads/HkrkeouXyx.png)
Note: Since every irreducible polynomial is seperable when $F$ has characteristic 0, a splitting field over $F$ always split a seperable polynomial. (Proposition 8.26(b))

## 9.8. The Primitive Element Theorem
- **Proposition 9.36**
![Screenshot 2024-11-30 at 11.23.55 PM](https://hackmd.io/_uploads/SyRZgou7ye.png)
- **Definition 9.37: Seperable Algebraic Extensions**
![Screenshot 2024-11-30 at 11.41.35 PM](https://hackmd.io/_uploads/rkZ4Vo_71g.png)
Note: 
    - Since minimal polynomials are irreducible, so if $F$ has characteristic 0, then minimal polynomials are seperable. (Proposition 8.26(b))
    - Inparticular,ifFhas characteristic 0, then every algebraic extension is separable
    - Clarify the the meaning of ["Primitive element"](https://en.wikipedia.org/wiki/Primitive_element_(finite_field))
- **Definition: Normal Algebraic Extensions** 
![Screenshot 2024-12-01 at 12.27.25 AM](https://hackmd.io/_uploads/BJr7Jnu7yl.png)
Note: This is definition from [wiki](https://en.wikipedia.org/wiki/Normal_extension). Definition 9.60 gives another definition of normal.
- **Theorem 9.39: Primitive Element Theorem**
![Screenshot 2024-12-01 at 12.34.20 AM](https://hackmd.io/_uploads/rkc5en_m1g.png)
![Screenshot 2024-12-01 at 12.35.42 AM](https://hackmd.io/_uploads/B1c1-3dmJg.png)
Note: [better explaination of Primitive Element Theorem](https://sites.math.washington.edu/~greenber/MATH404-PrimElem.pdf)
- **Lemma 9.41**
![Screenshot 2024-12-01 at 12.36.18 AM](https://hackmd.io/_uploads/BJEGb3_7yx.png)


## 9.9. Galois Extensions
- **Definition 9.42: Galois Extensions**
![Screenshot 2024-11-30 at 11.27.29 PM](https://hackmd.io/_uploads/B1hg-i_7kx.png)
![Screenshot 2024-12-01 at 12.37.35 AM](https://hackmd.io/_uploads/S1-IZhd7kg.png)
Note: 
    - **A Galois extension is an algebraic field extension that is normal and separable.**
    - Definition 9.42 is quite weird. Just follow the definition from [Wiki](https://en.wikipedia.org/wiki/Galois_extension).
- **Proposition 9.45: Galois groups as subgroups of permutation groups**
![Screenshot 2024-12-01 at 12.38.37 AM](https://hackmd.io/_uploads/BkHi-2O7Je.png)
- **Definition 9.46: Intermediate Fields**
![Screenshot 2024-12-01 at 12.39.46 AM](https://hackmd.io/_uploads/HJKCbh_71l.png)
- **Proposition 9.47**
![Screenshot 2024-12-01 at 12.41.20 AM](https://hackmd.io/_uploads/r1OEGh_Q1x.png)
![Screenshot 2024-12-01 at 12.41.55 AM](https://hackmd.io/_uploads/SJmUMhu7Jg.png)
Note: check example 9.50 (The splitting field of X4 − 2: fixed and intermediate fields).
- **Theorem 9.51**
![Screenshot 2024-12-01 at 12.42.59 AM](https://hackmd.io/_uploads/H1ncG3OQ1l.png)

## 9.10. The Fundamental Theorem of Galois Theory
- **Theorem 9.52: The Fundamental Theorem of Galois Theory**
![Screenshot 2024-12-01 at 12.45.22 AM](https://hackmd.io/_uploads/BJzQXnu7yx.png)
Note: check **9.10.1. The Galois Correspondence for $X^4 − 2$.**

## 9.11. Application: The Fundamental Theorem of Algebra
- **Proposition 9.53: A Version of the Intermediate Value Theorem**
![Screenshot 2024-12-01 at 12.47.57 AM](https://hackmd.io/_uploads/rJ40m3uXJg.png)
![Screenshot 2024-12-01 at 12.48.13 AM](https://hackmd.io/_uploads/ryEAm2uQ1g.png)
- **Theorem 9.54**
![Screenshot 2024-12-01 at 12.49.12 AM](https://hackmd.io/_uploads/B1JfNndXye.png)
- **Corollary 9.55: Fundamental Theorem of Algebra**
![Screenshot 2024-12-01 at 12.49.17 AM](https://hackmd.io/_uploads/SJ4GVhuQkl.png)

## 9.12. Galois Theory of Finite Fields
- **Definition 9.56: Field of order $p^d$:  $\mathbb{F}_{p^d}$**
![Screenshot 2024-12-01 at 1.18.22 AM](https://hackmd.io/_uploads/BJJxs3OXkx.png)
- **Theorem 9.57**
![Screenshot 2024-12-01 at 1.20.32 AM](https://hackmd.io/_uploads/BJOvsndXkx.png)
![Screenshot 2024-12-01 at 1.27.51 AM](https://hackmd.io/_uploads/HJvGa2u7Je.png)

## 9.13. A Plethora of Galois Equivalences
- **Lemma 9.59**
![Screenshot 2024-12-01 at 1.29.12 AM](https://hackmd.io/_uploads/HkC_phdQJe.png)
- **Definition 9.60: Normal Algebraic Extensions**
![Screenshot 2024-12-01 at 1.30.09 AM](https://hackmd.io/_uploads/HktoTh_7yg.png)
Note: In this book, only one property - "splits completetly in $K$" is used most of the time. The property - "has no roots in $K$" is seldom mentioned.
- **Theorem 9.61**
![Screenshot 2024-12-01 at 1.32.09 AM](https://hackmd.io/_uploads/HkR70humye.png)
![Screenshot 2024-12-01 at 1.32.29 AM](https://hackmd.io/_uploads/BJ0QA2u7Je.png)
- **Corollary 9.62**
![Screenshot 2024-12-01 at 1.42.21 AM](https://hackmd.io/_uploads/B10ueaOQye.png)

## 9.14. Cyclotomic Fields and Kummer Fields
In this section, we work exclusively with fields of characteristic 0.
- **Definition 9.63: Cyclotomic Fields**
![Screenshot 2024-12-01 at 1.51.14 AM](https://hackmd.io/_uploads/rJO9MaO7yl.png)
Note:
    - But be warned that some authors call $K$ a cyclotomic field if $K$ is contained in a splitting field of some $x^n − 1$. We also note that there is no need to use more than one polynomial, since a splitting field of $(x^n − 1)(x^m − 1)$ is the same as a splitting field of $x^{LCM(m,n)} − 1$; see Exercise 9.42
    - In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power n. [(wiki)](https://en.wikipedia.org/wiki/Root_of_unity)
- **Theorem 9.64**
![Screenshot 2024-12-01 at 9.37.08 PM](https://hackmd.io/_uploads/S1NK_At7yl.png)
![Screenshot 2024-12-01 at 9.40.29 PM](https://hackmd.io/_uploads/HkLUtCtXke.png)
Note$^{40}$: As our notation suggests, the values of κ(σ) depends only on σ; it does not depend on the choice of the primitive nth- root of unity ζ. See Exercise 9.43.
- **Theorem 9.66: Kronecker's Theorem**
![Screenshot 2024-12-01 at 9.42.14 PM](https://hackmd.io/_uploads/HyohtRYQkg.png)
- **Definition 9.67: Kummer Fields over $F$**
![Screenshot 2024-12-01 at 9.43.11 PM](https://hackmd.io/_uploads/ByE-c0F7kl.png)
Note: $F^*=F/\{0\}$
- **Theorem 9.68: Kummer Fields**
![Screenshot 2024-12-01 at 9.45.49 PM](https://hackmd.io/_uploads/Skxq5RYm1x.png)
Note: proof of (c)(1)
![Screenshot 2024-12-01 at 9.56.53 PM](https://hackmd.io/_uploads/rJBVa0KXJl.png)

## 9.15. Application: Insolubility of Polynomial Equations by Radicals
In this section, we work exclusively with fields of characteristic 0. 
P.S. It will be better to read the book for this section, since I skipped it.
- **Definition 9.70: Radical Extension**
![Screenshot 2024-12-01 at 9.58.20 PM](https://hackmd.io/_uploads/HJmtTAYQye.png)
Note$^{51}$: "Radical” is an old-fashioned synonym for an nth-root
- **Proposition 9.71**
![Screenshot 2024-12-01 at 9.58.49 PM](https://hackmd.io/_uploads/BJ6iaAYQ1g.png)
![Screenshot 2024-12-01 at 10.16.29 PM](https://hackmd.io/_uploads/SyeTbkcQJg.png)
- **Definition 9.73**
![Screenshot 2024-12-01 at 10.16.55 PM](https://hackmd.io/_uploads/SJ-gGk97Jx.png)
Note: Fields generated by constructible numbers are radical extensions, but they’re very special radical extensions in which only square roots are taken.
- **Remark 9.74**
![Screenshot 2024-12-01 at 10.32.21 PM](https://hackmd.io/_uploads/SJRqHy9Xkl.png)
![Screenshot 2024-12-01 at 10.32.36 PM](https://hackmd.io/_uploads/SyS5rJ5m1e.png)
- **Definition 9.75: Solvable Groups**
![Screenshot 2024-12-01 at 10.28.52 PM](https://hackmd.io/_uploads/SyasVy5QJx.png)
- **Lemma 9.76**
![Screenshot 2024-12-01 at 10.29.53 PM](https://hackmd.io/_uploads/SJKJSy9mkg.png)
- **Proposition 9.77**
![Screenshot 2024-12-01 at 10.30.44 PM](https://hackmd.io/_uploads/Byk7SycmJl.png)
- **Theorem 9.78**
![Screenshot 2024-12-01 at 10.34.03 PM](https://hackmd.io/_uploads/rkWJUy9myx.png)
- **Theorem 0.79**
![Screenshot 2024-12-01 at 10.36.58 PM](https://hackmd.io/_uploads/Hyq5L19Qyl.png)
- **Theorem 9.80**
![Screenshot 2024-12-01 at 10.36.28 PM](https://hackmd.io/_uploads/BJbuI1c71e.png)
- **Corollary 9.81**
![Screenshot 2024-12-01 at 10.39.54 PM](https://hackmd.io/_uploads/S1JrPkqmJg.png)
![Screenshot 2024-12-01 at 10.40.32 PM](https://hackmd.io/_uploads/S1fvw15X1g.png)
- **Lemma 9.83**
![Screenshot 2024-12-01 at 10.40.55 PM](https://hackmd.io/_uploads/HJ_Juy97kg.png)
- **Proposition 9.84**
![Screenshot 2024-12-01 at 10.41.01 PM](https://hackmd.io/_uploads/SkuJdyqQye.png)
- **Example 9.85: A Degree 5 Polynomial Not Solvable by Radicals**
![Screenshot 2024-12-01 at 10.41.16 PM](https://hackmd.io/_uploads/H1Kkuy5XJx.png)
Note: check the book for **9.15.3. Insolubility by Radicals: A Philosophical Digression.**

## 9.16. Linear Independence of Field Automorphisms
- **Theorem 9.87**
![Screenshot 2024-12-01 at 10.45.14 PM](https://hackmd.io/_uploads/HkSFu15Q1g.png)
- **Corollary 9.89: A Version of Hilbert’s Theorem 90**
![Screenshot 2024-12-01 at 10.45.24 PM](https://hackmd.io/_uploads/ryrYdyqXkg.png)