Extend the kronecker–weber theorem on abelian extensions of the rational numbers to any base number field

Description: This problem asks to extend the Kronecker-Weber Theorem on abelian extensions of the rational numbers to any base number field. The current Kronecker-Weber Theorem does this for the case of any imaginary quadratic field, showing that every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients.

Source: Wikipedia

Status: The work done by Shimura and Taniyama in the complex multiplication of abelian varieties gave rise to abelian extensions of CM-fields. Additionally, Stark’s conjecture resulted in a great conjectural development of L-functions, and is capable of producing concrete, numerical results. However, this problem still remains unresolved.

Hope this helped!!!

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

is only a line? i think so