In mathematics, **field arithmetic** is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.

Read more about Field Arithmetic: Fields With Finite Absolute Galois Groups, Fields That Are Defined By Their Absolute Galois Groups, Pseudo Algebraically Closed Fields

### Other articles related to "field arithmetic, field, fields, arithmetic":

**Field Arithmetic**- Pseudo Algebraically Closed Fields

... A pseudo algebraically closed

**field**(in short PAC) K is a

**field**satisfying the following geometric property ... Over PAC

**fields**there is a firm link between

**arithmetic**properties of the

**field**and group theoretic properties of its absolute Galois group ... nice theorem in this spirit connects Hilbertian

**fields**with ω-free

**fields**(K is ω-free if any embedding problem for K is properly solvable) ...

### Famous quotes containing the words arithmetic and/or field:

“I hope I may claim in the present work to have made it probable that the laws of *arithmetic* are analytic judgments and consequently a priori. *Arithmetic* thus becomes simply a development of logic, and every proposition of *arithmetic* a law of logic, albeit a derivative one. To apply *arithmetic* in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”

—Gottlob Frege (1848–1925)

“The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a *field* of force whose boundary conditions are experience.”

—Willard Van Orman Quine (b. 1908)