Quadratic Fields¶
A quadratic field is a specific type of number field—a field containing rational numbers and closed under basic algebraic operations. It extends the rational numbers by adding the square root of a non-square integer.
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It extends the rational numbers by adding the square root of a non-square integer.
The general representation of a quadratic field is \(\mathbb{Q}(\sqrt{d})\), where \(d\) is a non-square integer. In this field, numbers take the form \(a + b\sqrt{d}\), with \(a\) and \(b\) being rational numbers. The inclusion of the square root of \(d\) expands the field while preserving closure under its operations.
For instance:
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If \(d = 2\), then \(\mathbb{Q}(\sqrt{2})\) comprises numbers like \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational.
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If \(d = -1\), then \(\mathbb{Q}(\sqrt{-1})\) includes numbers of the form \(a + bi\), where \(a\) and \(b\) are rational, and \(i\) is the imaginary unit.
Quadratic fields find significant applications in number theory, algebraic number theory, and cryptography. Studying quadratic fields involves exploring their properties, ring of integers, units, and other associated algebraic structures.