Totally ramified prime

Definition

A totally ramified prime is a prime ideal in a number field that remains prime in an extension field and has a degree of ramification equal to the degree of the extension. This means that the prime ideal completely splits into powers of a single prime ideal in the extended ring of integers, indicating a strong connection between the two fields. Such primes play a key role in understanding how primes behave when moving between different number fields and can reveal insights about the structure of the field extensions.

5 Must Know Facts For Your Next Test

1. A totally ramified prime ideal corresponds to a unique prime ideal in the extension field, and all the primes above it are powers of this single ideal.
2. If a prime ideal is totally ramified, its ramification index will be equal to the degree of the extension.
3. For a totally ramified prime, the residue field at that prime is smaller than the residue field at its corresponding prime in the base field.
4. The concept of totally ramified primes is important in class field theory and algebraic number theory as it helps classify extensions based on how primes behave.
5. Examples of totally ramified primes are often found in cyclic extensions of number fields, where understanding their structure aids in analyzing more complex algebraic properties.

Review Questions

• How does a totally ramified prime relate to the behavior of primes when moving from a base field to an extension?
  • A totally ramified prime remains intact when moving to an extension field, meaning it does not decompose but instead fully ramifies. This means that if you start with a prime ideal in the base field, it will become a power of a single prime ideal in the extension. Understanding this behavior is crucial for studying how number fields interact and helps illustrate how primes can uniquely characterize extensions.
• Discuss the implications of a prime being totally ramified for its residue field and what this indicates about its relation to the base field.
  • When a prime is totally ramified, it results in its residue field being smaller than that of its corresponding prime in the base field. This indicates that there is no decomposition into multiple distinct residue classes. The shrinking of residue fields highlights how the structure of primes changes when extending fields and emphasizes the unique connection between primes across these fields.
• Evaluate the significance of totally ramified primes within class field theory and their impact on understanding Galois groups.
  • Totally ramified primes are significant in class field theory as they reveal how extensions can be classified based on their ramification properties. Their behavior informs us about Galois groups associated with extensions, where totally ramified primes contribute to understanding how subfields correspond to different subgroup structures. Analyzing these primes can lead to deeper insights into abelian extensions and help categorize fields based on their algebraic properties and symmetries.

Related terms

ramification: Ramification refers to the behavior of prime ideals in an extension of a number field, specifically how they split or remain intact, affecting their multiplicity.

decomposition: Decomposition describes how a prime ideal in one number field can factor into several prime ideals in an extension field, illustrating the relationship between these fields.

inertia degree: The inertia degree is the degree of the field extension associated with a prime ideal, indicating how the prime ideal behaves in terms of splitting or ramification.