Hopf-Galois module structure of degree p extensions of p-adic fields

Key Points

• This research aims to characterize when the ring of integers of a degree p extension of p-adic fields is free as a module over its associated order in the Hopf-Galois structure.
• Characterized conditions for freedom of the ring of integers in the context of Hopf-Galois modules.
• Examined degree p extensions of p-adic fields.
• Analyzed unique Hopf-Galois structures related to these fields.
• Fully characterized the condition for the ring of integers to be free as a module.
• Established the connection between the module structure and the unique Hopf-Galois structure.