The last post I made had a little terminology mix-up, but I liked reading the responses nevertheless.
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This is a tough one, I feel like the definitions of rings and fields are designed so that the rules, how one would calculate with whole numbers and reals, can be applied respectively.
The best I can do is maybe give you more tangible analogs of whole numbers and reals, that capture the spirit.
To visualize a ring, imagine a flat plane. You start of at some marked spot ‘the center’. And there is a grid of other points drawn on the floor. Periodically you get instructions to move. ‘4 to the east’, ‘2 to the north’ and sometimes something like ‘increase your distance to the center 4 times’
You find that these instructions always lead you to another grid point. If you need to increase the distance to the center you just find your location relative to it e.g. 2 dots west and 4 dots North, and repeat these steps the number of times that is specified.
This plane of grids together with those instructions forms an algebraic ring.
In contrast, if you also allowed instructions like ‘half the distance to the center’. The grid of points would not be enough for you to move around. Standing one dot away from the center, you couldn’t half the distance without leaving the grid. Instead you find that now you can basically move anywhere on the plane. You also cannot leave the plane. There’s no instruction telling you to go up. The set of instructions is again complete in that sense. The plane with those expanded instructions now represents an algebraic field.
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