Abstract Algebra, Lec 32: Fields of Order p^2, Vector Spaces, Q(sqrt(2)) Field Extension of Q

Abstract Algebra, Lecture 32. Field Theory and Galois Theory, Part 1.

(0:00) Field Theory and Galois Theory, Part 1.
(1:02) Our last main topic: in-depth field theory.
(4:56) Constructing fields of order p^2 (first note that Zp is, up to isomorphism, is the only field of order p when p is prime).
(6:43) A couple exercises from Chapter 17 to determine the number of reducible and irreducible monic quadratics there are over Zp, where p is prime (use PolynomialMod and Table on Mathematica to solve)...also note the formula for triangular numbers.
(15:11) Use should remember that Zp[x]/A is a field when A is the principal ideal generated by an irreducible polynomial over Zp (A will be a maximal ideal in Zp[x]).
(15:50) In this example, these fields have p^2 elements (5^2 = 25 when p = 5).
(19:55) Does the new field have a zero of f(x) (even though it did not have any zeros in Z5)? Yes, x + A is a zero.
(23:00) Z5[x]/A is a "field extension" of the field Z5 when Z5 is a subfield of Z5[x]/A (or at least an isomorphic image of Z5 is a subfield of Z5[x]/A).
(24:34) Definition of a vector space V over a field F (and the true nature of vector addition and scalar multiplication as binary operations (which satisfy closure)).
(30:00) Basic properties of vector spaces.
(32:07) Subspaces and subspace test.
(33:49) Kernel (null space) of a linear transformation and image (column space) of a linear transformation (can be determined using a matrix representation of the linear transformation with respect to the standard basis).
(36:51) Subspace spanned by a collection of vectors (set of all linear combinations of the vectors...this is also the smallest subspace of V containing these vectors).
(38:45) Example involving a 2 x 3 matrix.
(45:19) Linear dependence and linear independence of vectors in V over a field F.
(47:17) Basis and dimension of a vector space V over a field F (and invariance of basis size).
(49:23) Other worthwhile facts to know bout linear dependence/independence, bases, and coordinate vectors.
(51:08) Q(sqrt(2)) as a field extension of Q (also, Q(sqrt(2)) is a 2-dimensional vector space over Q with basis {1,sqrt(2)}).

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3:00 Not to mention it's widely used in Cryptography.