Simple Field Extension

Question 1

What is meant by algebraic?

Answer 1

Let K≤L be a field extension. Let b in L. We say b is algebraic over K if there is a non-zero polynomial p(x) in K[x] s.t. p(b) = 0.

Question 2

What is meant by simple algebraic extension?

What is meant by simple transcendental extension?

Answer 2

Let K≤L be a field extension. Let b in L.
If b is algebraic over K, then K(b) is a simple algebraic extension of K.
If b is transcendental (not algebraic) over K, then K(b) is a simple transcendental extension of K.

Question 3

When is b transcendental over K?

Answer 3

if and only if p(x) in K[x] with p(b) = 0 implies p(x) = 0.

Question 4

What is the minimal polynomial of b over K?

Answer 4

It’s the unique monic polynomial in K[x] of smallest possible degree with b as a root.

Question 5

Let K≤L be a field extension. What does [L:K] mean?

Answer 5

[L:K] = dim_K (L)

Question 6

Let K≤L be a field extension. What does [L:K] mean?

Answer 6

[L:K] = dim_K (L)