Konjugasi Kompleks sebagai Automorfisma
Definisi Fungsi
Diberikan fungsi \(\varphi: \mathbb{C} \to \mathbb{C}\) dengan definisi:
\[ \varphi(a + bi) = a - bi, \quad a, b \in \mathbb{R} \]
Ini adalah konjugasi kompleks, biasanya ditulis \(\varphi(z) = \overline{z}\).
Syarat Automorfisma Lapangan
Sebuah fungsi \(\varphi: \mathbb{C} \to \mathbb{C}\) disebut automorfisma lapangan jika:
- \(\varphi\) bijektif
- \(\varphi(x + y) = \varphi(x) + \varphi(y)\) untuk semua \(x, y \in \mathbb{C}\)
- \(\varphi(x \cdot y) = \varphi(x) \cdot \varphi(y)\) untuk semua \(x, y \in \mathbb{C}\)
- \(\varphi(1) = 1\)
Pembuktian
1. Homomorfisma Penjumlahan
Ambil \(z_1 = a + bi\), \(z_2 = c + di\) dengan \(a, b, c, d \in \mathbb{R}\).
\[ z_1 + z_2 = (a + c) + (b + d)i \]
\[ \varphi(z_1 + z_2) = (a + c) - (b + d)i \]
Di sisi lain:
\[ \varphi(z_1) + \varphi(z_2) = (a - bi) + (c - di) = (a + c) - (b + d)i \]
Terbukti \(\varphi(z_1 + z_2) = \varphi(z_1) + \varphi(z_2)\).
2. Homomorfisma Perkalian
\[ z_1 \cdot z_2 = (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]
\[ \varphi(z_1 \cdot z_2) = (ac - bd) - (ad + bc)i \]
Hitung \(\varphi(z_1) \cdot \varphi(z_2)\):
\[ \begin{aligned} \varphi(z_1) \cdot \varphi(z_2) &= (a - bi)(c - di) \\ &= ac - adi - bci + bd i^2 \\ &= ac - (ad + bc)i + bd(-1) \\ &= (ac - bd) - (ad + bc)i \end{aligned} \]
Terbukti \(\varphi(z_1 \cdot z_2) = \varphi(z_1) \cdot \varphi(z_2)\).
3. \(\varphi(1) = 1\)
\[ 1 = 1 + 0i \quad \Rightarrow \quad \varphi(1) = 1 - 0i = 1 \]
4. Bijektif
- Injektif: Jika \(\varphi(z_1) = \varphi(z_2)\), maka: \[ a - bi = c - di \ \Rightarrow\ a = c \text{ dan } b = d \ \Rightarrow\ z_1 = z_2 \]
- Surjektif: Ambil sembarang \(w = u + vi \in \mathbb{C}\). Pilih \(z = u - vi\), maka: \[ \varphi(z) = u + vi = w \]
Contoh Numerik
Contoh Penjumlahan
Misal \(z_1 = 1 + 2i\), \(z_2 = 3 + 4i\):
\[ z_1 + z_2 = 4 + 6i \quad \Rightarrow \quad \varphi(4 + 6i) = 4 - 6i \]
\[ \varphi(z_1) + \varphi(z_2) = (1 - 2i) + (3 - 4i) = 4 - 6i \]
Contoh Perkalian
\[ z_1 \cdot z_2 = (1 + 2i)(3 + 4i) = -5 + 10i \]
\[ \varphi(-5 + 10i) = -5 - 10i \]
\[ \varphi(z_1) \cdot \varphi(z_2) = (1 - 2i)(3 - 4i) = -5 - 10i \]
Kesimpulan
Fungsi \(\varphi(z) = \overline{z}\) memenuhi semua syarat automorfisma lapangan \(\mathbb{C}\):
\[ \boxed{\varphi \text{ adalah automorfisma lapangan } \mathbb{C}} \]