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3

Group theory has lots of applications in the sciences, such as to crystallography and quantum mechanics. Rings and fields basically don't, except in trivial ways like doing linear algebra over the complex numbers rather than the reals. I assume that's the reason why the curriculum is normally arranged the way it is, so that STEM students have the option of ...

6

I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses "Abstract Algebra I" and "Abstract Algebra II." I think there is lots of variation. Typically groups are in the first course, but Sylow theorems might be in the second course, along with rings. And the second ...

3

I think this certainly makes a lot more sense than teaching rings before groups. I think it might be a wash whether you teach all of the parts of groups and then all of the parts of rings and note the similarities along the way or if you collate them like this book does, but at least this seems pedagogically viable. I think my larger concern is how the ...

5

How about this: Let $k$ be an infinite field, and let $f \in k[x]$. Assume $f(t) = 0$ for all $t \in k$. Assume to the contrary that $f$ is not the zero polynomial. Then $f$ is a polynomial of degree $n \geq 1$. Choose distinct elements $z_1, z_2, z_3, \dots, z_{n+1}$ of $k$. By repeated use of the linear factor theorem, we know that \$f(x) = (x-z_1)(x-z_2)(...

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