What is hardy fields?

Hardy fields, also known as Hilbertian fields, were first introduced by the mathematician G.H. Hardy in the early 20th century. They are a class of fields with nice algebraic properties that are important in algebraic geometry and number theory.

Formally, a Hardy field is defined as an ordered field F that is closed under exponentiation and has a complete ordering such that every nonnegative element has a unique nth root for all positive integers n. In simpler terms, it means that the field has a well-behaved ordering and all kinds of exponential functions can be defined in them and manipulated algebraically.

Hardy fields are the basic objects used in the study of real closed fields and transcendental extensions. They have numerous applications in different mathematical fields, including algebraic geometry, number theory, and mathematical logic. They are particularly important in the study of transcendental numbers, as they give a framework for understanding their properties.

Overall, Hardy fields are a fundamental concept in modern mathematics, and their study has led to many important results and insights in various areas of mathematics.