Introduction: `"The Multiplicative Nature of Fields"`.
When we stack fields on top of one another , we create a "Tower". The fundamental law of these towers is that their dimensions (degrees) are multiplicative. If you know the "height" of each floor , you simply multiply them to find the height of the entire building.

Theory: `"The Generalized Tower Law"`.
1. The Tower Theorem: If `bbb"F"_1 subseteq bbb"F"_2 subseteq ... subseteq bbb"F"_r` is a sequence of finite extensions , then `[bbb"F"_r : bbb"F"_1] = [bbb"F"_r : bbb"F"_{r - 1}] * [bbb"F"_{r - 1} : bbb"F"_{r - 2}] * * * [bbb"F"_2 : bbb"F"_1]`.

2. The divisibility Rule: If `beta` is an element within the extension `bbb"F"(alpha)` , then the degree of `beta` over `bbb"F"` must be a divisor of the degree of `alpha` over `bbb"F"`.
`"L""ogic"`: Since `bbb"F" subseteq bbb"F"(beta) subseteq bbb"F"(alpha)` , the Tower Law dictates that `[bbb"F"(alpha) : bbb"F"] = [bbb"F"(alpha) : bbb"F"(beta)][bbb"F"(beta) : bbb"F"]`.

`"The Overview Tower Connection"`:
- Top Level: `bbb"F"(alpha)` (Dimension `n` over `bbb"F"`).
- Middle Level: `bbb"F"(beta)` (Dimension `m` over `bbb"F"`).
- Bottom Level: `bbb"F"` (The Base Field).
- Requirement: `m` must be a factor of `n`.

Exercise: `"The Impossible Root"`.
Problem: Prove that no element in `QQ(sqrt 2)` can be a root of the polynomial `p(x) = x^3 - 2`.

Solution: The degree of `alpha = sqrt 2` over `QQ` is `2` (from `x^2 - 2)`.
- Let `beta` be a root of `x^3 - 2`. The degree of `beta` over `QQ` is `3`.
- If `beta` where in `QQ(sqrt 2)` , then by the Divisibility Rule , `deg(beta , QQ)` would have to divide `deg(sqrt 2 , QQ`).
- Since `3` does not divide `2` , it is impossible for `beta` to exist within that field. QED