---
# System prepended metadata

title: 'Fields [ch.5]'
tags: [abstract algebra]

---

# Fields [ch.5]

## 5.1. Introduction to Fields
- **Definition 5.1**
![Screenshot 2024-11-25 at 5.27.15 PM](https://hackmd.io/_uploads/HyNWH2Zm1x.png)
![Screenshot 2024-11-25 at 5.27.51 PM](https://hackmd.io/_uploads/B1dzS2-Xyl.png)

## 5.2. Abstract Fields and Homomorphisms
- **Definition 5.3: Unit Group of $R$, $F^*$**
![Screenshot 2024-11-25 at 5.29.53 PM](https://hackmd.io/_uploads/HJmiH3Z71x.png)
- **Proposition 5.4**
![Screenshot 2024-11-25 at 5.30.41 PM](https://hackmd.io/_uploads/SJMpBn-Xyg.png)
Note: As above mentioned, every map between fields is injective.

## 5.3. Interesting Examples of Fields
- **Example 5.8**
![Screenshot 2024-11-25 at 5.35.31 PM](https://hackmd.io/_uploads/By_ewhbmJx.png)
- **Example 5.9: Finite Field $F_p$**
![Screenshot 2024-11-25 at 5.33.21 PM](https://hackmd.io/_uploads/Bksv82bmJe.png)
- **Example 5.10: Skew Fields, Division Rings**
![Screenshot 2024-11-25 at 5.33.46 PM](https://hackmd.io/_uploads/HJJtInWQye.png)
Note: check other examples in textbooks.

## 5.4. Subfields and Extension Fields
- **Definition 5.11: Subfields**
![Screenshot 2024-11-25 at 5.34.56 PM](https://hackmd.io/_uploads/S1R7wh-Q1l.png)
- **Definition 5.12: Extension Fields**
![Screenshot 2024-11-25 at 5.38.49 PM](https://hackmd.io/_uploads/BJCTD3W7ye.png)
![Screenshot 2024-11-25 at 5.39.25 PM](https://hackmd.io/_uploads/ryRpv3-7Jg.png)
Note$^5$: Note that K/F is just a piece of convenient notation. It does not mean that we are taking the quotient of K by F , despite the fact that it’s similar to the notation R/I that we use for the quotient of a ring by an ideal.
- **Proposition 5.15**
![Screenshot 2024-11-25 at 5.41.06 PM](https://hackmd.io/_uploads/rJQEuhZ7Jl.png)
- **Definition 5.16: Degree of $K$ over $F$**
![Screenshot 2024-11-25 at 5.43.09 PM](https://hackmd.io/_uploads/H1K6OhZX1x.png)
![Screenshot 2024-11-25 at 5.53.15 PM](https://hackmd.io/_uploads/B1gzs2WXke.png)
Note: 
    - The extension field $L/K$ has $[L:K] = 1$, means L = K
    - The degree of $K/F$ can be seen as the minimum number of bases picked out in $K$ such that the bases in $K$ with scalars in $F$ can build entire $K$.
    - A field $F_{p^n}$ is an field extension of $F_p$, and $[F_{p^n}:F_P]=n$ 
        - example: $F_2=\{0,1\},F_{2^2}=\{0,1,\alpha, \alpha+1\}$
- **Theorem 5.18: Multiplicativity of Degree in Towers of Fields**
![Screenshot 2024-11-25 at 5.51.43 PM](https://hackmd.io/_uploads/SkgCchWmJl.png)

## 5.5. Polynomial Rings
- **Definition 5.19: Polynomial Rings**
![Screenshot 2024-11-25 at 5.54.48 PM](https://hackmd.io/_uploads/ByKPin-m1g.png)
Note: $f(x)\in F[x]$ is a polynomial ring with scalars in a field $F$.
- **Proposition 5.20: Division-with-Remainder for Polynomials**
![Screenshot 2024-11-25 at 5.55.49 PM](https://hackmd.io/_uploads/r1Bso3-mkx.png)
- **Theorem 5.21**
![Screenshot 2024-11-25 at 5.58.05 PM](https://hackmd.io/_uploads/r1fE32ZQ1e.png)

## 5.6. Building Extension Fields
- **Definition 5.22: Irreducible Polynomials**
![Screenshot 2024-11-25 at 5.59.56 PM](https://hackmd.io/_uploads/H1Cc2nb7Jg.png)
![Screenshot 2024-11-25 at 6.03.30 PM](https://hackmd.io/_uploads/BkBup2WQkl.png)
![Screenshot 2024-11-25 at 6.04.19 PM](https://hackmd.io/_uploads/r15s63WXkg.png)
Note: 
    - Linear factors like $x-1, x+3, ...$, are always irreducible and are the only kinds of irreducible polynomials have roots in $F$.
- **Proposition 5.26** 
![Screenshot 2024-11-25 at 6.04.52 PM](https://hackmd.io/_uploads/HyAApnbXke.png)
Note: 
    - $f(x)F[x]$ is an ideal of a ring $F[x]$ which is generated by $f(x)$.
    - $F[x]/f(x)F[x]$ means a ring $F[x]$ quotiented by a polynomial $f(x)$. Thinking a quotient reing like $\mathbb{Z}/m\mathbb{Z}$
- **Theorem 5.27**
![Screenshot 2024-11-25 at 6.13.15 PM](https://hackmd.io/_uploads/r1snyTWXJg.png)
![Screenshot 2024-11-25 at 6.15.00 PM](https://hackmd.io/_uploads/SJVXg6-XJg.png)
Note: 
    - The second picture is the proof of (c).
    - In (c), the root is not a value in $F$ but a polynomial in $K_f$. 

## 5.7. Finite Fields
- **Proposition 5.28**
![Screenshot 2024-11-25 at 6.31.03 PM](https://hackmd.io/_uploads/B1PyETW71l.png)
- **Theorem 5.29**
![Screenshot 2024-11-25 at 6.31.41 PM](https://hackmd.io/_uploads/SyBfVTZmkl.png)
- **Theorem 5.31**
![Screenshot 2024-11-25 at 6.32.34 PM](https://hackmd.io/_uploads/HytSNpbX1l.png)
![Screenshot 2024-11-25 at 6.35.09 PM](https://hackmd.io/_uploads/SkIJSabmJe.png)
![Screenshot 2024-11-25 at 6.35.46 PM](https://hackmd.io/_uploads/S1G-SpWXJx.png)
Note:
    - **Any finite field's order is always a prime number  or a power of a prime number.** ([proof](htthttps://www.reddit.com/r/learnmath/comments/1dm1sr4/why_is_the_order_of_a_finite_field_a_prime_number/ps://))
    - The multiplicative group of every finite field is cyclic. (O is omitted)
## Others
- **Minimal Polynomials [(wiki)](https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory))**
![Screenshot 2024-11-30 at 1.26.35 AM](https://hackmd.io/_uploads/SJhSiPvmJg.png)
Note: A minimal polynomial is irreducible.
- field $F_{p^n}$ is an field extension of $F_p$, and $[F_{p^n}:F_P]=n$ 
    - example: $F_2=\{0,1\},F_{2^2}=\{0,1,\alpha, \alpha+1\}$
- [Zp and Fp difference](htthttps://crypto.stackexchange.com/questions/12065/what-is-the-difference-between-and-security-of-z-p-and-f-pps://)
