# Fields [ch.5]

## 5.1. Introduction to Fields
- **Definition 5.1**

## 5.2. Abstract Fields and Homomorphisms
- **Definition 5.3: Unit Group of $R$, $F^*$**
- **Proposition 5.4**
Note: As above mentioned, every map between fields is injective.

## 5.3. Interesting Examples of Fields
- **Example 5.8**
- **Example 5.9: Finite Field $F_p$**
- **Example 5.10: Skew Fields, Division Rings**
Note: check other examples in textbooks.

## 5.4. Subfields and Extension Fields
- **Definition 5.11: Subfields**
- **Definition 5.12: Extension Fields**
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**
- **Definition 5.16: Degree of $K$ over $F$**
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**

## 5.5. Polynomial Rings
- **Definition 5.19: Polynomial Rings**
Note: $f(x)\in F[x]$ is a polynomial ring with scalars in a field $F$.
- **Proposition 5.20: Division-with-Remainder for Polynomials**
- **Theorem 5.21**

## 5.6. Building Extension Fields
- **Definition 5.22: Irreducible Polynomials**
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**
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**
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**
- **Theorem 5.29**
- **Theorem 5.31**
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))**
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://)