---
# System prepended metadata

title: 'Fields [ch8.]'
tags: [abstract algebra]

---

# Fields [ch8.]

## 8.1. Algebraic Numbers and Transcendental Numbers
- **Definition 8.1: Algebraic, Transcendental Numbers**
![Screenshot 2024-11-25 at 6.39.34 PM](https://hackmd.io/_uploads/H1neL6WQkg.png)
![Screenshot 2024-11-27 at 10.16.38 AM](https://hackmd.io/_uploads/H1D-QgEmkl.png)
![Screenshot 2024-11-27 at 10.16.16 AM](https://hackmd.io/_uploads/SyZxmlN7kg.png)
- **Definition 8.4: Smallest subring and subfield contains $F$ and $\alpha$**
![Screenshot 2024-11-27 at 10.14.19 AM](https://hackmd.io/_uploads/Hy2_fg4Xyg.png)
- **Theorem 8.5**
![Screenshot 2024-11-27 at 10.13.14 AM](https://hackmd.io/_uploads/BygBGe4myl.png)
- **Proposition 8.6**
![Screenshot 2024-11-27 at 10.18.25 AM](https://hackmd.io/_uploads/SJpuQeVX1g.png)
![Screenshot 2024-11-27 at 10.19.05 AM](https://hackmd.io/_uploads/BJK57gVX1e.png)
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**
![Screenshot 2024-11-27 at 10.21.36 AM](https://hackmd.io/_uploads/Hk6VNxEXkl.png)
![Screenshot 2024-11-27 at 10.22.19 AM](https://hackmd.io/_uploads/HJiUExNQ1g.png)
- **Corollary 8.10**
![Screenshot 2024-11-27 at 10.22.35 AM](https://hackmd.io/_uploads/r1w_NxN71g.png)
- **Lemma 8.11**
![Screenshot 2024-11-27 at 10.23.12 AM](https://hackmd.io/_uploads/By_c4gEX1x.png)
![Screenshot 2024-11-27 at 10.24.01 AM](https://hackmd.io/_uploads/rJGaExVXke.png)

## 8.3. Splitting Fields, Separability, and Irreducibility
- **Definition 8.13: Splitting fields**
![Screenshot 2024-11-27 at 10.24.49 AM](https://hackmd.io/_uploads/H1ierlE7yl.png)
![Screenshot 2024-11-27 at 10.26.00 AM](https://hackmd.io/_uploads/rk_EreEmkl.png)
- **Theorem 8.17**
![Screenshot 2024-11-27 at 10.26.29 AM](https://hackmd.io/_uploads/B1DKHgNXyg.png)
- **Definition 8.18: (Formal) Derivatives**
![Screenshot 2024-11-27 at 10.27.49 AM](https://hackmd.io/_uploads/rJJ6Hx4X1x.png)
- **Proposition 8.19**
![Screenshot 2024-11-27 at 10.28.51 AM](https://hackmd.io/_uploads/B1O1Lg4Q1x.png)
![Screenshot 2024-11-27 at 10.30.27 AM](https://hackmd.io/_uploads/ByBSIxEXJl.png)
- **Definition 8.21: Separable**
![Screenshot 2024-11-27 at 10.50.10 AM](https://hackmd.io/_uploads/SkylsgE7ke.png)
- **Proposition 8.24**
![Screenshot 2024-11-27 at 10.51.14 AM](https://hackmd.io/_uploads/SJGQigV71e.png)
![Screenshot 2024-11-27 at 10.53.14 AM](https://hackmd.io/_uploads/HyT9oeNQ1e.png)
![Screenshot 2024-11-27 at 11.00.16 AM](https://hackmd.io/_uploads/BkQHpxNX1x.png)
- **Theorem 8.26**
![Screenshot 2024-11-27 at 11.00.40 AM](https://hackmd.io/_uploads/BJGvTeNQJl.png)
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**
![Screenshot 2024-11-28 at 2.02.34 PM](https://hackmd.io/_uploads/SyiFYOrmJx.png)
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 $f$**
![Screenshot 2024-11-28 at 2.06.32 PM](https://hackmd.io/_uploads/BJZ_9uBmJe.png)
- **Lemma 8.30**
![Screenshot 2024-11-28 at 2.07.33 PM](https://hackmd.io/_uploads/H1Fs9_B7ke.png)
- **Theorem 8.31: Gauss's Lemma**
![Screenshot 2024-11-28 at 2.08.51 PM](https://hackmd.io/_uploads/SyIbourXJl.png)
- **Corollary 8.32: Irreducibiity via Reduction mod p**
![Screenshot 2024-11-28 at 2.11.39 PM](https://hackmd.io/_uploads/rkqjidH7yx.png)
![Screenshot 2024-11-28 at 2.13.14 PM](https://hackmd.io/_uploads/SkrZ3dB71e.png)
![Screenshot 2024-11-28 at 2.13.22 PM](https://hackmd.io/_uploads/HJHb2_BQ1g.png)
- **Corollary 8.35: Eisenstein Irreducibility Criterion**
![Screenshot 2024-11-28 at 2.13.45 PM](https://hackmd.io/_uploads/ryBmndB7Je.png)
- **Example 8.36: Cyclotomic Polynomials**
![Screenshot 2024-11-28 at 2.16.21 PM](https://hackmd.io/_uploads/By1ThuS7Jx.png)
- **Definition 8.37: nth Cyclotomic Polynomials**
![Screenshot 2024-11-28 at 2.17.49 PM](https://hackmd.io/_uploads/Sk4MadSQJl.png)
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**
![Screenshot 2024-11-28 at 2.25.29 PM](https://hackmd.io/_uploads/HJge1YSm1g.png)

## 8.6. Ruler and Compass Constructions
- Skipped