This is the field of Gaussian rational numbers.
Normalized defining polynomial
\( x^{2} + 1 \)
Invariants
| Degree: | $2$ |
| |
| Signature: | $[0, 1]$ |
| |
| Discriminant: |
\(-4\)
\(\medspace = -\,2^{2}\)
|
| |
| Root discriminant: | \(2.00\) |
| |
| Galois root discriminant: | $2\approx 2.0$ | ||
| Ramified primes: |
\(2\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4=2^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4}(1,·)$, $\chi_{4}(3,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $0$ |
| |
| Torsion generator: |
\( a \)
(order $4$)
|
| |
| Regulator: | \( 1 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr =\mathstrut &\frac{2^{0}\cdot(2\pi)^{1}\cdot 1 \cdot 1}{4\cdot\sqrt{4}}\cr\approx \mathstrut & 0.785398163397448 \end{aligned}\]
Galois group
| A cyclic group of order 2 |
| The 2 conjugacy class representatives for $C_2$ |
| Character table for $C_2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *2 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *2 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.
Spectrum of ring of integers
Additional information
The ring of integers, $\Z[i]$, is a Euclidean domain, hence unique factorization domain, with norm $$N(a+bi)=a^2+b^2 = (a+bi)(a-bi).$$ As a result, it is connected to the question of which positive integers can be written as the sum of two squares, and more specifically, to the theorem of Fermat that a prime number $p$ can be written as the sum of two squares if and only if $p\not\equiv 3\pmod 4$, and that if $p=a^2+b^2$, then the representation is unique subject to $0<a\leq b$.