Numbers of Type Rational a plus b root 2 form Field
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Theorem
Real Numbers of Type Rational $a + b \sqrt 2$ form Field
Let $\Q \sqbrk {\sqrt 2}$ denote the set:
- $\Q \sqbrk {\sqrt 2} := \set {x \in \R: x = a + b \sqrt 2: a, b \in \Q}$
that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers.
Then the algebraic structure:
- $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$
where $+$ and $\times$ are conventional addition and multiplication on real numbers, is a field.
Complex Numbers of Type Rational $a + b \sqrt 2$ form Field
Let $\Q \sqbrk {\sqrt 2}$ denote the set:
- $\Q \sqbrk {\sqrt 2} := \set {x \in \C: x = a + b \sqrt 2: a, b \in \Q}$
that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers.
Then the algebraic structure:
- $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$
where $+$ and $\times$ are conventional addition and multiplication on real numbers, is a number field.