Gaussian integer

What to expect can be seen already for the Gaussian integers.

If is an odd prime number, then we have in the Gaussian integers.

Its double cover acts on a 28-dimensional lattice over the Gaussian integers.

This is equivalent to determining the number of Gaussian integers with norm less than a given value.

The period lattices are of a very special form, being proportional to the Gaussian integers.

Consider again the case of the Gaussian integers.

The conjecture is not valid over the larger domain of Gaussian integers.

For example, the Gaussian integers form a lattice in C.

In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.

Any such complex torus has the Gaussian integers as endomorphism ring.

The Gaussian integers are a special case of the quadratic integers.

It is easy to see graphically that every complex number is within units of a Gaussian integer.

Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers.

In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers.

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.

The Gaussian integers are an integral domain because they are a subring of the complex numbers.

The Gaussian integers form a ring, as do the Eisenstein integers.

Important examples include the Gaussian integers and the Eisenstein integers.

Thus these primes (and 2) occur as norms of Gaussian integers, while other primes do not.

Many of the other applications of the Euclidean algorithm carry over to Gaussian integers.

A Gaussian integer is a Gaussian prime if and only if either:

These numbers are now called the ring of Gaussian integers, denoted by Z[i].

In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas.

Every number in , e. g. the Gaussian integers , is uniquely representable as a finite code, possibly with a sign.

Gaussian integers: those complex numbers where both and are integers are also quadratic integers.