Irreducibility in multiple dimensions

In summary, when dealing with multidimensional abstract algebra, we can use analogues of Gauss's lemma and Eisenstein's criterion for any polynomial ring R[x] over a UFD R. This involves pairing up powers of X with powers of Y and treating the powers of Y as coefficients of X. We can then apply the same steps as in the standard irreducibility criterion, although we may need to consider degenerate cases due to the division by the ideal (xy).
  • #1
Simfish
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So we know some of the irreducibility criteria when we have one dimension X.

But what about multidimensional abstract algebra?

From Dummit Foote, we get that we can pair up every power of X with the powers of Y that happen to be associated with that power of X, and then treat those powers of Y as coefficients of X. Do we then use the same steps that we use in our standard irreducibility criterion? (other than the division by the ideal (xy) - which can lead to degenerate cases?)
 
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  • #2
There are analogues of Gauss's lemma and Eisenstein's criterion for any polynomial ring R[x] over a UFD R. (In particular, for R[x,y]=(R[x])[y].) I'm not sure if this is what you're asking though.
 

Related to Irreducibility in multiple dimensions

1. What is irreducibility in multiple dimensions?

Irreducibility in multiple dimensions refers to the idea that a system or phenomenon cannot be simplified or broken down into smaller components in order to understand or explain it. In other words, it is a concept that describes the complexity and interconnectivity of various factors and variables within a multi-dimensional system.

2. Why is irreducibility important in scientific research?

Irreducibility is important in scientific research because it acknowledges the interconnectedness and complexity of natural phenomena. By recognizing that a system cannot be reduced to its individual parts, scientists are able to approach problems and questions with a more holistic perspective. This can lead to a deeper understanding of the underlying mechanisms and relationships within a system.

3. How is irreducibility in multiple dimensions different from reductionism?

Irreducibility in multiple dimensions is different from reductionism in that reductionism seeks to break down complex systems into simpler components in order to understand them. Irreducibility, on the other hand, acknowledges that certain systems cannot be reduced to their individual parts and must be studied as a whole.

4. Can irreducibility be applied to all scientific disciplines?

Yes, irreducibility can be applied to all scientific disciplines. Whether it is studying the human brain, ecosystems, or the laws of physics, there are always multiple dimensions and factors at play that cannot be reduced to a single explanation. Irreducibility helps scientists approach problems and questions from a more comprehensive perspective.

5. How does irreducibility impact the development of scientific theories?

Irreducibility can have a significant impact on the development of scientific theories. By acknowledging the complexity and interconnectedness of natural phenomena, scientists may need to revise or expand existing theories in order to account for all dimensions and factors. This can lead to a more complete and accurate understanding of the natural world.

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