Prove there are exactly 4 non-isomorphic algebras among algebras Af

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In summary: Again, all elements of Af can be written as a + bi. However, the operations of addition and multiplication in this case are slightly different:(a + bi) + (c + di) = (a + c) + (b + d)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i + (bd - ad)(g+h+k)In summary, we have shown that there are exactly 4 non-isomorphic algebras among algebras Af, which correspond to the 4 possible cases for the roots of f. I hope this has helped you understand the problem better.
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snazy20
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Homework Statement


Let f = ax^3+bx^2+cx+d in R[x] be a cubic polynomial with real coefficients (a not equal to 0). Let g,h,k in Complex Numbers be roots of f. Let Af = R[x]/(f).

Prove that there are exactly 4 non-isomorphic algebras among algebras Af.

The Attempt at a Solution



Proved f either has 1 real and 2 distinct complex nonreal roots or 3 real roots. In the former case some of roots can coincide (3 or 2 or none).

From here I have become stumped, completely. Hope I have given enough for some help - please let me know if anything is unclear, but I have no idea where to go.

Thanks in advance
 
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Hello,

Thank you for your post. It seems like you have made some good progress in your attempt at a solution. I will try to guide you through the remaining steps.

First, let's consider the case where f has 3 real roots. In this case, we can write f as:

f = a(x-g)(x-h)(x-k)

where a is a nonzero real coefficient and g, h, and k are the three real roots of f. Now, let's consider the algebra Af. Since f is a polynomial with real coefficients, all elements of Af can be written as a + bi, where a and b are real numbers and i is the imaginary unit. We can then define the operations of addition and multiplication in Af as follows:

(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Now, let's consider the case where f has 1 real and 2 distinct complex nonreal roots. In this case, we can write f as:

f = a(x-g)(x-h)^2

where a is a nonzero real coefficient, g is the real root of f, and h is a complex nonreal root of f. Again, all elements of Af can be written as a + bi. However, the operations of addition and multiplication in this case are slightly different:

(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (ac - bd) + (ad + bc)i + (bd - ad)h

Now, let's consider the case where f has 2 real and 1 complex nonreal root. In this case, we can write f as:

f = a(x-g)(x-h)(x-k)

where a is a nonzero real coefficient, g and h are the two real roots of f, and k is a complex nonreal root of f. Again, all elements of Af can be written as a + bi. The operations of addition and multiplication in this case are the same as in the first case.

Finally, let's consider the case where f has 3 distinct complex nonreal roots. In this case, we can write f as:

f = a(x-g)(x-h)(x-k)

where a is a nonzero real
 

What is an algebra?

An algebra is a mathematical structure consisting of a set of elements, operations, and axioms that define how the elements can be combined and manipulated.

What does "non-isomorphic" mean in relation to algebras?

Two algebras are said to be non-isomorphic if they cannot be transformed into each other through a bijective mapping that preserves the algebraic structure.

How can it be proven that there are exactly 4 non-isomorphic algebras among algebras Af?

This can be proven by first defining the set of algebras Af and then systematically examining all possible combinations and structures within this set to determine the number of non-isomorphic algebras.

What makes an algebra unique and non-isomorphic from others in the set Af?

An algebra can be uniquely identified by its defining characteristics, such as the set of elements, operations, and axioms. If any of these characteristics differ, the algebra is considered to be non-isomorphic from others in the set.

How is the concept of isomorphism important in the study of algebras?

Isomorphism is important in the study of algebras as it helps to identify and distinguish between different algebraic structures. It allows for a deeper understanding of the relationships and properties within a set of algebras.

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