Find Quotient Field of Gaussian Integers

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In summary, Gaussian integers are complex numbers of the form a + bi, where a and b are integers and i is the imaginary unit. The quotient field of Gaussian integers is the field of fractions formed by taking all possible ratios of Gaussian integers, denoted by ℚ(i). To find this field, we find the inverse of each Gaussian integer and simplify the resulting fractions. This field is important in number theory and allows us to extend the set of integers to include complex numbers. It is also related to other fields such as the field of complex numbers, algebraic numbers, and rational numbers.
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Find the quotient field of a ring of Gaussian integers?
 
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Well, this sounds like homework... so what have you tried on this problem?

Quotient field? Do you mean the field of fractions?
 

What are 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.

What is the quotient field of Gaussian integers?

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).

How do you find the quotient field of Gaussian integers?

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.

What is the importance of finding the quotient field of Gaussian integers?

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.

How is the quotient field of Gaussian integers related to other fields?

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.

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