# Fields [ch8.]

## 8.1. Algebraic Numbers and Transcendental Numbers
- **Definition 8.1: Algebraic, Transcendental Numbers**
- **Definition 8.4: Smallest subring and subfield contains $F$ and $\alpha$**
- **Theorem 8.5**
- **Proposition 8.6**
Note:
- $f(x)$ should be the minimal polynomial of $\alpha$.
- [How to find a minimal polynomial of $\alpha$](https://www.youtube.com/watch?v=kFjOqReUHJo)

## 8.2. Polynomial Roots and Multiplicative Subgroups
- **Theorem 8.8**
- **Corollary 8.10**
- **Lemma 8.11**

## 8.3. Splitting Fields, Separability, and Irreducibility
- **Definition 8.13: Splitting fields**
- **Theorem 8.17**
- **Definition 8.18: (Formal) Derivatives**
- **Proposition 8.19**
- **Definition 8.21: Separable**
- **Proposition 8.24**
- **Theorem 8.26**
Note: (b) means there is a field extension $K/F$, so that $f(x)$ factors seperablely in $K$

## 8.4. Finite Fields Revisited
- **Theorem 8.28**
Note:
- let $F_p = Z/pZ$ has characteristic $P$
- All subfields have the same characteristic as the original field, so $L/F_p$ has the same characteristic.
- **Theorem 8.29**

## 8.5. Gauss’s Lemma, Eisenstein’s Irreducibility Criterion, and Cyclotomic Polynomials
- **Definition 8.29: Content of $f$**
- **Lemma 8.30**
- **Theorem 8.31: Gauss's Lemma**
- **Corollary 8.32: Irreducibiity via Reduction mod p**
- **Corollary 8.35: Eisenstein Irreducibility Criterion**
- **Example 8.36: Cyclotomic Polynomials**
- **Definition 8.37: nth Cyclotomic Polynomials**
Note:
- For example, we have noted that although the cyclotomic polynomials $\Phi_n(x)$ are irreducible!
- Highly used in lattice-based cryptography
- **Example 8.38: Some Polynomials that Are Probably Irreducible**

## 8.6. Ruler and Compass Constructions
- Skipped