The document defines key terms and provides examples related to field extensions in abstract algebra. It defines the degree of a field extension as the dimension of the field as a vector space over the base field. The degree of the complex field over the real field is given as an example. An algebraic element is defined as an element of a field extension that is a root of a polynomial with coefficients in the base field. Square root of 2 and 1/5 are given as examples of algebraic elements over the rational field. It also shows that the square root of 2 plus the square root of 3 is an algebraic element over the rational field by constructing a polynomial of which it is a root.

1. A Presntation on
Question no 10
Present By: Raj Koirala
Firast semester Batch 2020
Kathmandu University

2. What do you mean by the degree of field
extension ?
Define algebraic element with an example.
Show that {2 + (3 )}is a algebraic over Q.

3. Degree of field extension
The degree of field K over F is the
dimension of K as a vector space over F.
The degree of K over F is denoted by
[K:F] if [K:F] is finite then we say K is
finite extension of field F.
Fore example:
The degree of C over R. i.e. [C:R]=2
because the set {1, i} forms a basis of C
over R.

4. Definition of algebraic element
Let K/ F be a field extension. Then the
element aє K is said to be algebraic over F
if every element of K is algebraic over F.
Fore example:
(a) 2 є R is algebraic over Q since
P(x)=x²-2 є Q[x] and P(x) =(2 )² -2=0
(b) 1/5 є R is algebraic over Q since
P(x)=5x-1 є Q[x] and p(x)= 5*1/5=0.

5. Show that {2 + (3 )}is a algebraic over Q.
Let K/ F be a field extension. Then the
element aє K is said to be algebraic over F
if every element of K is algebraic over F.
Fore example:
(a) 2 є R is algebraic over Q. since
P(x)=x²-2 є Q[x] and P(x) =(2 )² -2=0
(b) 1/5 є R is algebraic over Q since
P(x)=5x-1 є Q[x] and p(x)= 5*1/5-1=0.

6. Thank You