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Find the quotient field of a ring of Gaussian integers?
Gaussian integers are complex numbers of the form a + bi, where a and b are both integers and i is the imaginary unit (√-1). They are named after mathematician Carl Friedrich Gauss.
The quotient field of Gaussian integers is the field of fractions formed by taking all possible ratios of Gaussian integers. In other words, it is the set of all possible fractions of the form (a + bi)/(c + di), where a, b, c, and d are integers and i is the imaginary unit. This field is denoted by ℚ(i).
To find the quotient field of Gaussian integers, we first need to find the inverse of each Gaussian integer. This can be done by taking the complex conjugate of the Gaussian integer and dividing it by the square of its norm. Then, we can use these inverses to form all possible fractions and simplify them to their simplest form, resulting in the quotient field.
Finding the quotient field of Gaussian integers is important in number theory and algebra, as it allows us to extend the set of integers to include complex numbers. This field is also useful in solving certain mathematical problems, such as finding solutions to polynomial equations with complex coefficients.
The quotient field of Gaussian integers is a subfield of the field of complex numbers, as all Gaussian integers are also complex numbers. It is also a subfield of the field of algebraic numbers, as all Gaussian integers are algebraic numbers. Additionally, the quotient field of Gaussian integers is isomorphic to the field of rational numbers.