Fields

Question 1

Basis?

Answer 1

B is a basis of V over F if the elements in B are linearly independent over F and every element of V can be written as a linear combination of those in B.

Question 2

Extension field?

Answer 2

E is an extension field of F if F subseteq E and the operations of F are those of E restricted to F.

Question 3

Fundamental theorem of field theory?

Answer 3

Let F be a field and f(x) be a nonconstant polynomial in F[x]. There there is an extension field of F in which f(x) has a zero.

Question 4

Splitting field of f(x)?

Answer 4

The smallest extension of F in which f(x) splits. Always exists.

Question 5

Vector space?

Answer 5

A set V over a field F such that V is an Abelian group under addition and for each a in F and v in V, av is in V. Also, for all a,b in F and u,v in V:
a(v+u)=av+au
(a+b)v=av+bv 
a(bv)=(ab)v
1v=v

Question 6

Explain F(a) isomorphic to F[x]/<p></p>

Answer 6

If p(x) is irreducible over F and a is a zero of p(x) in some extension E of F, then it holds. Furthermore, if deg(p(x)=a, then the powers of a form a basis of F(a).

Question 7

When does a polynomial f(x) over F have a multiple zero in some extension E?

Answer 7

If and only if f(x) and f’(x) have a common factor of positive degree in F[x].

Question 8

Perfect field?

Answer 8

A field is perfect if it has characteristic 0, or if F has characteristic p and F^p = {a^p | a in F} = F.

Question 9

What can be said about the zeros of an irreducible over a splitting field?

Answer 9

They all have the same multiplicity.

Question 10

Algebraic element?

Answer 10

a is algebraic over F if a is the zero of some nonconstant polynomial in F[x].

Question 11

Algebraic extension?

Answer 11

E is an algebraic extension of F if every element of E is algebraic over F.

Question 12

Minimal polynomial?

Answer 12

If a is algebraic over F, then there is a unique monic irreducible p(x) in F[x] such that p(a)=0.

Question 13

Finite extension?

Answer 13

[E:F]

Question 14

[E:F] = ?

Answer 14

If K is an extension of F and E is an extension of K, then [E:F] = [E:K][K:F]

Question 15

Are finite extensions algebraic?

Answer 15

Yes. Consider an extension E of F such that [E:F]=n. Then take a in E. Then {1,a,a^2,…,a^n} is linearly dependent over F…

Question 16

What is the order of a finite field?

Answer 16

Prime-power.

Question 17

GF(p^n)?

Answer 17

Up to isomorphism there is a unique field of order p^n, GF(p^n) is this field.

Question 18

As a group under addition, GF(p^n) is isomorphic to ?

Under multiplication?

Answer 18

Z_p x Z_p x … x Z_p (x=\oplus).

GF(p^n) \ {0} is isomorphic to Z_{p^n-1} and thus is cyclic.

Question 19

[GF(p^n):GF(p)] = ?

Answer 19

n

Question 20

What are the subfields of a finite field?

Answer 20

For each divisor m of n, GF(p^n) has a unique subfield of order p^m. These are the only subfields.