Mapping on Quadratic Integers over 2 to Conjugate is Automorphism
Theorem
Let $\Z \sqbrk {\sqrt 2}$ denote the set of quadratic integers over $2$:
- $\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$
that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers.
Then the mapping $\phi: \Z \sqbrk {\sqrt 2} \to \Z \sqbrk {\sqrt 2}$ defined as:
- $\forall x = a + b \sqrt 2 \in \Z \sqbrk {\sqrt 2}: \map \phi {a + b \sqrt 2} = a - b \sqrt 2$
is a ring automorphism.
Proof
We have Quadratic Integers over 2 form Subdomain of Reals.
First we note that:
- $\forall x \in \Z \sqbrk {\sqrt 2}: \map \phi x \in \Z \sqbrk {\sqrt 2}$
Proof of Bijection
Let $\map \phi {a + b \sqrt 2} = \map \phi {a' + b' \sqrt 2}$.
Then:
- $a - b \sqrt 2 = a' - b' \sqrt 2$
and so:
- $a + b \sqrt 2 = a' + b' \sqrt 2$
So $\phi$ is injective.
Now let $y = c + d \sqrt 2 \in \Z \sqbrk {\sqrt 2}$.
We have that:
\(\ds \map \phi {c + \paren {-d} \sqrt 2}\) | \(=\) | \(\ds c - \paren {-d} \sqrt 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c + d \sqrt 2\) |
Hence:
- $\forall y \in \Z \sqbrk {\sqrt 2}: \exists x \in \Z \sqbrk {\sqrt 2}: \map \phi x = y$
and so $\phi$ is surjective.
So $\phi$ is a bijection.
Proof of Morphism
Now consider $x = a + b \sqrt 2, y = c + d \sqrt 2$.
\(\ds \map \phi x + \map \phi y\) | \(=\) | \(\ds \map \phi {a + b \sqrt 2} + \map \phi {c + d \sqrt 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a - b \sqrt 2} + \paren {c - d \sqrt 2}\) | Definition of $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a + c} - \paren {b + d} \sqrt 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {\paren {a + c} + \paren {b + d} \sqrt 2}\) | Definition of $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {x + y}\) |
Then:
\(\ds \map \phi x \, \map \phi y\) | \(=\) | \(\ds \map \phi {a + b \sqrt 2} \, \map \phi {c + d \sqrt 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a - b \sqrt 2} \paren {c - d \sqrt 2}\) | Definition of $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a c + 2 b d} - \paren {b c + a d} \sqrt 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {\paren {a c + 2 b d} + \paren {b c + a d} \sqrt 2}\) | Definition of $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {x y}\) |
So $\phi$ is a ring homomorphism which is also a bijection, whose image equals its domain.
Hence the result by definition of ring automorphism.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms: Example $44$