Absolute ramification refers to a specific type of ramification in field extensions where the prime ideal's height is equal to its degree. This concept is crucial in understanding the behavior of prime ideals when passing from one field to its extension, especially in the context of algebraic number theory. Absolute ramification gives insight into how primes split, remain inert, or ramify completely, affecting the structure of the extension and its associated valuations.
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In absolute ramification, every prime ideal in the base field corresponds to a unique prime ideal in the extension that completely ramifies.
An example of absolute ramification occurs in extensions of local fields, where every prime of the base field has a single generator in the extension.
The degree of an extension can be determined by examining the number of times a prime ideal appears in the factorization after extending.
Absolute ramification can lead to interesting results regarding unit groups and class groups in algebraic number theory.
Understanding absolute ramification helps in determining properties such as completeness and valuation within a given extension.
Review Questions
How does absolute ramification affect the splitting behavior of prime ideals in field extensions?
Absolute ramification affects prime ideals by ensuring that each prime ideal from the base field corresponds to a unique prime ideal in the extension that is fully ramified. This means that instead of splitting into multiple factors or remaining inert, the prime ideal will contribute to one prime ideal in the extended field. Understanding this behavior is essential for grasping how algebraic structures behave under various extensions.
What is the relationship between absolute ramification and the concepts of ramification index and inertia degree?
Absolute ramification is closely related to both ramification index and inertia degree. In cases of absolute ramification, the ramification index will equal the degree of the field extension since all primes completely ramify. The inertia degree will typically be 1, as there are no new residue classes introduced at this stage. Together, these concepts provide a complete picture of how primes behave within extensions.
Evaluate the implications of absolute ramification on the structure and properties of local fields and their extensions.
The implications of absolute ramification on local fields are significant, as it affects both their structure and properties. When an extension exhibits absolute ramification, it often leads to well-defined unit groups and class groups, which are crucial for understanding local field behavior. Moreover, since every prime fully ramifies, this condition contributes to key attributes like completeness and uniformity within the valuation ring, ultimately influencing how algebraic number theorists approach problems involving these fields.
Related terms
ramification index: The ramification index measures how much a prime ideal in a base field is 'spread out' in an extension, indicating how many times it appears in the factorization of prime ideals.
inertia degree: The inertia degree is the degree of the field extension over a residue field, reflecting how many different primes are represented in the residue class after extension.
discriminant: The discriminant is a polynomial that provides important information about the roots of a polynomial equation and helps determine the behavior of primes in field extensions.