I'm reading a linear algebra book (Linear Algebra by Georgi E. Shilov, Dover Books) and the very start of the book discusses fields. 9 field axioms discussing addition and multiplication are given then the author goes on to discuss common sets of numbers.
The integers are identified as being a set of numbers which is not a field because there does not exist a reciprocal element for every integer (axiom # 8 in this book states the existence of a reciprocal element $B$ for a number $A$ such that $AB=1$). The author goes on to call the real numbers a field, and asserts that an axiomatic treatment can be had by supplementing the field axioms with the order axioms and the least upper bound axiom.
My understanding consists of the following statements I believe are facts:
- zero is a member of the reals
- there exists no reciprocal element of zero that is a real number
Given those two facts it seems to me that the reals fail the same test for being a field that the author states the integers fail. Yet the author is calling the real numbers a field. To my mind this is a contradiction.
Is there a resolution to this apparent contradiction? I'm a total beginner at this sort of math (I'm an engineer by training, not a mathematician!) and would appreciate any assistance!