Algebraic Relations of Basic Sets

[figure: ℙ; ℕ; ℤ; ℚ; 𝔸; ℝ; ℂ; ℤ/nℤ; ℤ_{p}; 𝔽_{p^{m}}; ℚ_{p}; ⋃_{m=1}^{∞} 𝔽_{p^{m!}}; 𝔸_{p}; \overline{ℚ_{p}}; \overline{𝔸_{p}}; indecomposible; set; commutative; semi-group; commutative; ring with 1; field; algebraic; closed field; topologic; closed; “natural”; archimedic; finite; p-adic; topologic; closed]

ℕ = {1, 2, 3, 4, 5, …} 1 ∈ ℕ, n ∈ ℕ ⇒ n + 1 ∈ ℕ

ℙ = {2, 3, 5, 7, 11 …} ℙ ⊂ ℕ, ∀ p ∈ ℙ ∀ n, m ∈ ℕ , n·m = p : n ∈ {1} ∪^{.} {p}

ℤ = {…, −2, −1, 0, 1, 2, …} ℕ ⊂ ℤ, ∀ k ∈ ℤ ∃ n, m ∈ ℕ : k + m = n

ℤ/nℤ = {0, 1, 2, 3, …, n−1} (n ∈ ℕ) ∀ a ∈ ℤ ∃! x ∈ ℤ/nℤ ∃ b ∈ ℤ : a = b·n + x

ℤ_{p} = {∑_{i=0}^{∞} x_{i}p^{i} : x_{i} ∈ ℤ/pℤ} (p ∈ ℙ)

ℚ = \{\frac{a}{n} : a ∈ ℤ n ∈ ℕ\} \frac{a}{1} = a ∈ ℤ \frac{a}{n} = \frac{b}{m} ⇔ a·m = b·n

ℚ_{p} = {∑_{i=k}^{∞} x_{i}p^{i} : k ∈ ℤ x_{i} ∈ ℤ/pℤ} (p ∈ ℙ)