CH 2 - Field Extensions

Question 1

Define a field extension and a base field.

Define the degree of K over F.

Answer 1

Let F and K be fields such that F is a subfield of K. We then say that K is an extension of F. We also call F the base field of the extension.

Question 2

State the Tower Law.

Answer 2

Moreover, in the setting of the theorem, if {v1, v2, . . . , vm} is a basis for L over K and {w1, w2, . . . , wn} is a basis for K over F, then { viwj | 1<= i<= m, 1<= j<= n } is a basis for L over F.

Question 3

Define an algebraic element over F.

Define an algebraic extension of F.

State and prove a lemma on finite vs algebraic extensions.

Answer 3

Question 4

Define the smallest subfield of K that contains both F and the elements α1, α2, . . . , αn.

Define a simple extension of F.

Answer 4

Question 5

Define a minimum polynomial of α over F.

State a 5-part theorem on properties of the minimum polynomial f of α.

Answer 5

Question 6

State a theorem on a simple extension F(α) over F and the degree of α over F

State a corollary on forming a basis for the simple extension F(α) over F.

Answer 6

Question 7

State a theorem on when K is a finite extension of F.

State and prove a corollary on linear combinations of algebraic elements over F.

Answer 7

Question 8

State a theorem on existence of a simple extension F(α) of F.

Answer 8