Algebraic conjugates are the roots of a polynomial equation with rational coefficients that share the same minimal polynomial. They play a crucial role in understanding the structure of number fields, particularly in relation to the norm and trace functions. By exploring these conjugates, we can better analyze the behavior of algebraic numbers and their relationships within a number field.
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Algebraic conjugates are significant in determining the structure of extensions in number theory, especially when studying Galois groups.
If an algebraic number $\eta$ is a root of a polynomial $p(x)$, then all other roots of $p(x)$ are considered its algebraic conjugates.
The norm of an algebraic number is equal to the product of its algebraic conjugates, which can be used to compute field degrees.
The trace is simply the sum of all algebraic conjugates, providing insights into various algebraic properties like discriminants.
Algebraic conjugates can help identify whether an algebraic number is integral over its base field, leading to broader implications in number theory.
Review Questions
How do algebraic conjugates relate to the minimal polynomial of an algebraic number?
Algebraic conjugates are directly tied to the minimal polynomial of an algebraic number because they are all roots of this polynomial. If you have a polynomial with rational coefficients, all roots (or solutions) including the original number and its conjugates must satisfy this polynomial equation. Therefore, understanding the minimal polynomial allows us to find and work with these conjugates effectively.
Discuss how the norm and trace of an algebraic number are influenced by its algebraic conjugates.
The norm and trace functions derive directly from the properties of algebraic conjugates. The norm is defined as the product of all the conjugates, while the trace is defined as their sum. This means that by knowing one or both of these values, we can infer information about the corresponding set of conjugates. For example, if we know the trace of an algebraic number, we can determine how many conjugates there are and what their relationships might be.
Evaluate the implications of algebraic conjugates in determining whether an algebraic number is integral over its base field.
Algebraic conjugates have significant implications for assessing if an algebraic number is integral over its base field. If all the conjugates of an algebraic number are also integral, then that number itself is considered integral. This property is crucial for understanding ring structures within number fields and has further consequences on factorization and ideal theory in algebraic number theory.
Related terms
Minimal Polynomial: The monic polynomial of least degree that has a given algebraic number as a root and whose coefficients are in the base field.
Norm: A function that associates a non-negative integer to an algebraic number, representing the product of its conjugates.
Trace: The sum of the conjugates of an algebraic number, which provides important information about its algebraic properties.