Complex field

Abstract and algebraic definitions

Existence of the field can be established by means of various models. One model is to regard the complex field as the set ⁠${\displaystyle \mathbb {R} ^{2}}$⁠ equipped with addition and multiplication laws on ordered pairs ⁠${\displaystyle (a,b)}$⁠ and ⁠${\displaystyle (a',b')}$⁠: $${\displaystyle (a,b)+(a',b')=(a+a',b+b'),\quad (a,b)(a',b')=(aa'-bb',ab'+a'b).}$$ In this model the element ⁠${\displaystyle i}$⁠ corresponds to the element ⁠${\displaystyle (0,1)}$⁠, and the real numbers are embedded as the set of pairs ⁠${\displaystyle (a,0)}$⁠. Another is the standard construction of a splitting field using a quotient of the polynomial ring⁠${\displaystyle \mathbb {R} [X]}$⁠, given below.