Fields

Question 1

What is a field

Answer 1

A field F is a commutative ring in which every non-zero element has a multiplicative inverse.

Question 2

What are elements of a vector space called

Answer 2

Vectors

Question 3

If K≤L are fields. Then what can be deduced in relevance of vector spaces

Answer 3

L is a vector space over K

Question 4

Let L be a field and let K≤L.
What do we call K?
And what is an Extension Field?

Answer 4

K is a subfield of L, if K is a field using the same operations of + and x as in L, and it’s only a subfield if and only if the criterion can be applied.

Question 5

Subfield Criterion

Answer 5

1 in K,
a-b in K, whenever a,b in K,
ab in K, whenever a,b in K,
a^-1 in K whenever a ≠ 0 in K

Question 6

If K is a subfield of (R) or (C), what field is always contained in it? And prove it

Answer 6

(Q)

Question 7

What is meant by a field generated by something?

Answer 7

Let X be a subset of the field L. The subfield of L

Question 8

What is meant by a field generated by something?

Answer 8

Let X be a subset of the field L. The subfield of L generated by X is defined to be the smallest subfield of L which contains X.

Question 9

How do you show that, K, say, is the subfield generated by X?

Answer 9

K is a subfield and X≤K, and

if J is a subfield containing X, then K≤J

Question 10

what is meant by K(Y) when K is a subfield of the field L, and Y≤L.

Answer 10

K(Y) stands for the subfield of L generated by K U Y. K(Y) is called the field extension of K obtained by adjoining Y.

Question 11

How do we prove that, given a subfield, K, of a field L, and a,b,c,d,e, K(a,b) = K(c,d,e)?

Answer 11

1. Show that a,b in K(c,d,e). This shows that K(c,d,e) is a subfield of L which contains K, a, b. Therefore,
   K(a,b) ≤ K(c,d,e), by the minimality of the field K(a,b).
2. Show that c,d,e in K(a,b). This would show that
   K(c,d,e) ≤ K(a,b), by the minimality of the field K(c,d,e).

Question 12

How do we prove that, given a subfield, K, of a field L, and a,b,c,d,e, K(a,b) = K(c,d,e)?

Answer 12

1. Show that a,b in K(c,d,e). This shows that K(c,d,e) is a subfield of L which contains K, a, b. Therefore,
   K(a,b) ≤ K(c,d,e), by the minimality of the field K(a,b).
2. Show that c,d,e in K(a,b). This would show that
   K(c,d,e) ≤ K(a,b), by the minimality of the field K(c,d,e).

Question 13

What is a simple field extension?

Answer 13

A field extension L of a field K is said to be simple if L = K(a) for some a in L, that is, if L can be obtained from K by adjoining a single element.