Algebraic number field Updated 2025-07-16
The set of all algebraic numbers forms a field.
This field contains all of the rational numbers, but it is a quadratically closed field.
Like the rationals, this field also has the same cardinality as the natural numbers, because we can specify and enumerate each of its members by a fixed number of integers from the polynomial equation that defines them. So it is a bit like the rationals, but we use potentially arbitrary numbers of integers to specify each number (polynomial coefficients + index of which root we are talking about) instead of just always two as for the rationals.
Each algebraic number also has a degree associated to it, i.e. the degree of the polynomial used to define it.
Algebraic structure Updated 2025-07-16
A set plus any number of functions , such that each satisfies some properties of choice.
Key examples:
  • group: one function
  • field: two functions
  • ring: also two functions, but with less restrictive properties
Alibaba Updated 2025-07-16
A linear map is a (1,1) tensor Updated 2025-07-16
A linear map can be seen as a (1,1) tensor because:
is a number, . is a dual vector, and W is a vector. Furthermoe, is linear in both and . All of this makes fullfill the definition of a (1,1) tensor.
Aluminium Updated 2025-07-16
Alla Rakha Updated 2025-07-16
Video 1.
Tabla Solo in Jhaptal by Alla Rakha, featuring Ravi Shankar (2003)
Source.

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