# Galois Theory [ch.9]

## 9.1. What Is Galois Theory?
- **Definition 9.1: Algebraic Extension**
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**
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**
Note$^2$: The converse of Corollary 9.5 is false; see Exercise 9.7.
- **Theorem 9.6**
- **Definition 9.8: Minimal Polynomial of $\alpha$ over $F$**
Note$^4$: We are using the fact that every ideal in F [x] is principal; see Theorem 5.21.
- **Theorem 9.10**

## 9.4. Algebraically Closed Fields
- **Definition 9.11: Algebraically Closed**
Note: For example, the field $\mathbb{C}$ is algebraically closed.
- **Proposition 9.12**
- **Definition 9.14: Algebraic Closures**
- **Theorem 9.15**

## 9.5. Automorphisms of Fields
- **Example 9.18**
- **Definition 9.19: Galois Groups**
- **Proposition 9.20**
- **Proposition 9.22**

## 9.6. Splitting Fields — Part 1
- **Lemma 9.23**
- **Theorem 9.25**
- **Corollary 9.26**
- **Corollary 9.27**
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**
- **Lemma 9.31**
- **Proposition 9.34**
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**
- **Definition 9.37: Seperable Algebraic Extensions**
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**
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**
Note: [better explaination of Primitive Element Theorem](https://sites.math.washington.edu/~greenber/MATH404-PrimElem.pdf)
- **Lemma 9.41**

## 9.9. Galois Extensions
- **Definition 9.42: Galois Extensions**
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**
- **Definition 9.46: Intermediate Fields**
- **Proposition 9.47**
Note: check example 9.50 (The splitting field of X4 − 2: fixed and intermediate fields).
- **Theorem 9.51**

## 9.10. The Fundamental Theorem of Galois Theory
- **Theorem 9.52: The Fundamental Theorem of Galois Theory**
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**
- **Theorem 9.54**
- **Corollary 9.55: Fundamental Theorem of Algebra**

## 9.12. Galois Theory of Finite Fields
- **Definition 9.56: Field of order $p^d$: $\mathbb{F}_{p^d}$**
- **Theorem 9.57**

## 9.13. A Plethora of Galois Equivalences
- **Lemma 9.59**
- **Definition 9.60: Normal Algebraic Extensions**
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**
- **Corollary 9.62**

## 9.14. Cyclotomic Fields and Kummer Fields
In this section, we work exclusively with fields of characteristic 0.
- **Definition 9.63: Cyclotomic Fields**
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**
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**
- **Definition 9.67: Kummer Fields over $F$**
Note: $F^*=F/\{0\}$
- **Theorem 9.68: Kummer Fields**
Note: proof of (c)(1)

## 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**
Note$^{51}$: "Radical” is an old-fashioned synonym for an nth-root
- **Proposition 9.71**
- **Definition 9.73**
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**
- **Definition 9.75: Solvable Groups**
- **Lemma 9.76**
- **Proposition 9.77**
- **Theorem 9.78**
- **Theorem 0.79**
- **Theorem 9.80**
- **Corollary 9.81**
- **Lemma 9.83**
- **Proposition 9.84**
- **Example 9.85: A Degree 5 Polynomial Not Solvable by Radicals**
Note: check the book for **9.15.3. Insolubility by Radicals: A Philosophical Digression.**

## 9.16. Linear Independence of Field Automorphisms
- **Theorem 9.87**
- **Corollary 9.89: A Version of Hilbert’s Theorem 90**