# Which totally complex number field's embeddings correspond to geometric rotations/reflections in an argand diagram?

For which totally complex number fields $K$ with embeddings $\{ \sigma_1, \dots, \sigma_m\}$ do we have the equality:

$$|\sigma_1(x)| = |\sigma_2(x)| = \dots = |\sigma_m(x)|,$$

for all $x \in K$ where $|\cdot|$ corresponds to the complex absolute value $|x| = (x\bar{x})^{1/2}$? In other words, which number fields have embeddings that do not affect the distance of a coordinate on an argand diagram?

• What do the units have to be? They would also have to have absolute value 1. Apr 4 '18 at 21:51
• @TCiur be that as it may, how does it help answer my question? Apr 4 '18 at 22:03
• @TCiur I believe the quadratic extension $\mathbb{Q}(\sqrt{d})$ where $d$ is a squarefree negative integer satisfies the condition too. Apr 4 '18 at 22:18
• Your condition requires that all archimedean places of K coincide, but this does not occur. The only possibility is when there is only 1 archimedean place, and this happens when we have a pair of complex embeddings, I.E an imaginary quadratic field Apr 4 '18 at 23:53
• I might be showing my ignorance here, but don't they all satisfy that condition? Apr 5 '18 at 3:29

$|x| = |\sigma(x)| \implies \sigma(x) = u \cdot x$ for some complex unit $u$.
Assume $\sigma(x) = u \cdot x$, then $\sigma(x+1) = u\cdot x + 1 = w\cdot x + w$ for some other unit $w$. This would imply that $|x+1| = |u\cdot x + 1|$. This is absurd unless $u\cdot x = \bar{x}$ or $x$. Hence the galois group must consist only of the identity and the complex conjugate (only imaginary quadratic fields fit the condition).
• In any cyclotomic field, non-complex-conjugate galois actions on $\zeta + \zeta^{-1}$ will change the absolute value (think geometrically). Apr 5 '18 at 16:36