n Java,
Create a class called Complex for performing arithmetic with
complex numbers. Complex numbers have the form
realPart + imaginaryPart * i
where i is square root of -1
Use floating-point variables to represent the private data of
the class. Provide a constructor that enables an object of this
class to be initialized when it is declared.
Provide a no-argument constructor with default values in case no
initializers are provided. Provide public methods that perform the
following operations:
a)...
Let E be an extension field of a finite field F, where F has q
elements. Let a in E be an element which is algebraic over F with
degree n. Show that F(a) has q^n elements. Please provide an unique
answer and motivate all steps carefully. I also prefer that the
solution is provided as written notes.
a)Adoubledata field(private)named realfor real part of a complex
number.
b)Adoubledata field(private)named imgfor imaginarypart of a
complex number.
c)A no-arg constructor that creates a default complex number
with real 0and img 0.
d)Auser-defined constructorthat creates a complex number with
given 2 numbers.
e)The accessor and mutator functions for realand img.
f)A constant function named addition(Complex&comp1,
Complex&comp2) that returns the sum of two givencomplex
numbers.
g)Aconstantfunction named subtraction(Complex&comp1,
Complex&comp2) that returns the subtractionof two givencomplex
numbers.
h)A constant function named multiplication(Complex&comp1,
Complex&comp2)...
Problem 3. Let F ⊆ E be a field extension.
(i) Suppose α ∈ E is algebraic of odd degree over F. Prove that
F(α) = F(α^2 ). Hints: look at the tower of extensions F ⊆ F(α^2 )
⊆ F(α) and their degrees.
(ii) Let S be a (possibly infinite) subset of E. Assume that
every element of S is algebraic over F. Prove that F(S) = F[S]
Theorem: Let K/F be a field extension and let a ∈ K be algebraic
over F. If deg(mF,a(x)) = n, then
1. F[a] = F(a).
2. [F(a) : F] = n, and
3. {1, a, a2 , ..., an−1} is a basis for F(a).