Abstract Algebra is more rightly considered meta-mathematics than mathematics proper, because it can be used to describe the structures that exist within mathematics from a general standpoint. The basic notions of Groups, Rings, Fields, and Algebraic

Extensions provide a framework from which to examine almost all of mathematics.

These notions serve as unifying concepts that interlace such seemingly disparate subjects as geometry, analysis, number theory, topology and even applied mathematics. Nicomedes Alonso III

Abstract Algebra

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SOLVABILITY BY RADICALS - Linear Equation

Clearly the root of the linear equation ax + b = 0

(1)

is given in terms of the coefﬁcients a and b by x = −b/a as long as a = 0.

Nicomedes Alonso III

Abstract Algebra

2 of 29

SOLVABILITY BY RADICALS - Quadratic Equation

We know that the roots of the quadratic equation, ax 2 + bx + c = 0

(2)

are given by the well-known quadratic formula

x=

−b ±

b2 − 4ac

,

2a

a=0

in terms of a, b and c.

Nicomedes Alonso III

Abstract Algebra

3 of 29

SOLVABILITY BY RADICALS - Cubic Equation

For the general cubic polynomial equation

3

2

ax + bx + cx + d = 0,

(3)

the roots are given by Cardan’s formulas p =

q

=

R

=

c a d

p3

27

P

=

Q

=

3

3a2 bc −

a

3

b2

−

3a2

−

2b3

27a3

q2

+

−

+

4

q

2

q

2

√

R

+

−

√

R

and with cube roots chosen properly, the roots of (3) are given by

P+Q−

2

ωP + ω Q −

2

ω P + ωQ −

b

3a

b

3a

b

3a

where ω = 1 is a cube root of 1.

Nicomedes Alonso III

Abstract Algebra

4 of 29

SOLVABILITY BY RADICALS - Quartic Equation (a)

The general quartic equation ax 4 + bx 3 + cx 2 + dx + e = 0,

(4)

b 3 c d e x = − x2 − x − . a a a a

(5)

may be rewritten as x4 +

Adding

b2 2 x 4a2

to both sides of (5) we obtain

Nicomedes Alonso