Rational Numbers form Subfield of Complex Numbers
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Theorem
Let $\struct {\Q, +, \times}$ denote the field of rational numbers.
Let $\struct {\C, +, \times}$ denote the field of complex numbers.
$\struct {\Q, +, \times}$ is a subfield of $\struct {\C, +, \times}$.
Proof
From Rational Numbers form Subfield of Real Numbers, $\struct {\Q, +, \times}$ is a subfield of $\struct {\R, + \times}$.
From Real Numbers form Subfield of Complex Numbers, $\struct {\R, +, \times}$ is a subfield of $\struct {\C, + \times}$.
Thus from Subfield of Subfield is Subfield $\struct {\Q, +, \times}$ is a subfield of $\struct {\C, + \times}$.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 16$. Subfields: Example $21$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 88$