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An appendix in my linear algebra textbook gives a brief introduction into fields. It then gives the two following examples:

  1. $1 + 1 = 0$

  2. Neither the set of positive integers nor the set of integers with the usual definitions of addition and multiplication is a field, for in either case $a + c = 0$ and $bd = 1$ does not hold, where $b$ is a nonzero element.

All of the examples preceding these two were sensible in conventional algebra. However, these two examples seem absurd.

I would greatly appreciate it if someone could please explain what the textbook is saying here.

Thank you.

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    https://en.wikipedia.org/wiki/Finite_field – 2017-01-04
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    @SpamIAm Thank you. Your article says, "As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules"; is this the same as making up your own axioms? – 2017-01-04
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    @ThePointer No, not your own axioms; finite fields follow the same axioms as other fields. Which axioms for fields in general does your textbook introduce? – 2017-01-04
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    Sure, just like vector spaces, there is a set of axioms that must be satisfied in order to be a field. (See [here](https://en.wikipedia.org/wiki/Field_(mathematics)#Definition_and_illustration).) They are things you would expect, like commutativity of addition and multiplication. – 2017-01-04
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    @HagenvonEitzen My textbook only mentions "fields" -- not "finite fields" or anything else. – 2017-01-04
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    @SpamIAm Interesting. Thank you very much for your assistance. – 2017-01-04
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    Is that passage verbatim? Because it is terribly written for a textbook (or any source trying to be even a little precise). – 2017-01-04

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These two examples are not absurd.

The first shows that fields as defined are not what you might first think. Here integers modulo $2$ - i.e. the numbers $0$ and $1$ with appropriate definitions of addition and multiplication, form a field with $1+1=0$. It is a field of characteristic $2$. Set theory provides a notion of Boolean Algebra which also has $1+1=0$ (and some of fields with $1+1=0$ are infinite). The integers taken modulo a prime $p$ form a field of characteristic $p$ where adding $p$ copies of $1$ gives zero. The rational numbers and real numbers are two fields of characteristic zero (you can't add any number of $1$s and get to zero) - such fields are necessarily infinite.

The second, by contrast, shows that some familiar and useful algebraic objects are not fields. The positive integers miss out on subtraction (no answer for $1-2$) and the integers still fail on division ($\frac 12$ is not available).

The examples are there to help you to know what a field is and is not beyond what your first thoughts may be, and to prepare you to meet unfamiliar examples in future work.