__Definition__: what something is.

1: thing.

+: association between things.

__Equality__: a=b means a and b represent the same thing, and is true or false depending on what they represent. 1+1=0 and 1=12 are false in Z, but 1+1=0 in Z2 and 1ft=12in are true.

“Advanced” abstract mathematics is a matter of definitions abstracted from familiar concepts and usually presented to students without reference to the familiar concepts.

An

__Exercise in definition__from Birkhoff and McLane, definition of Field assumed, which you can just squeak through comfortably if you are familiar with the concept of a polynomial. Otherwise you can still piece it together, but you won’t feel comfortable with it.

Definition: Let K be any field, and F any subfield of K. An element of c of K will be called algebraic over F if c satisfies a polynomial equation with coefficients not all zero over F.

An element c of K which is not algebraic over F is called transcendental.

Theorem: If c is transcendental over F, the subfield F(c) generated by F and c is isomorphic to the field F(x) of all rational forms in an indeterminate x, with coefficients in F. The isomorphism may be so chosen that c <-> x and a <-> a, for each a in F.

Definition: Polynomial Form in x over F: P(x) = a0+a1x+a2x2

^{+}….anx

^{n}.

Definition: Rational Polynomial Form: P(x)/Q(x).

Teachers don’t emphasize definitions. That is why many students don’t “get” math. The teacher says something that is not intuitive and the student doesn’t understand it. It’a a definition. It’a a definition. It’a a definition. If intuitive, it should be explained as such.

Let ICC = Ice Cream Cone. There is nothing to understand. IT’S A DEFINITION.