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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers , form an integral domain , usually written as Z [ i ] {\displaystyle \mathbf {Z} [i]} or Z [ i ] . {\displaystyle \mathbb {Z} [i].} [ 1 ]
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.
In particular, the integers (also see Fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs. If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD ...
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
The ordinary integers and the Gaussian integers allow a division with remainder or Euclidean division. For positive integers N and D, there is always a quotient Q and a nonnegative remainder R such that N = QD + R where R < D. For complex or Gaussian integers N = a + ib and D = c + id, with the norm N(D) > 0, there always exist Q = p + iq and R ...
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, [4]: §§69–71 Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z[i], the ring of Gaussian integers.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."