More class group questions

Dear Sir,

In reference to your material posted online for “A few class group”,

In the 2nd example where m(x)= x^3+4x-2. How did you factorise m(x) over F7? How did you do it? Why did u choose to divide m(x) with (x+5) initially? Is there a general way for obtaining the factors of m(x)? Is it by trial and error?

Also from the 3rd example, I am really confused by the sentence “Thus, we see that {e,[P2],[P3],[P2P3]} is an order 4 subgroup of CLk in which each element has order 2.” What does this mean? What is e? and why do we consider a subgroup of order 4? Where did the P7 go? How did you derive the inequalities for |Clk|?

After the 3rd example I seem to be really confused in determining if Pi is a principal or not. Can you please help in giving a general way to decide if each prime is a principle idea? Does it differ for every question? Is the prime a principal as long as the N(pi)=(pi)?

Thank you. Thanks for holding up this blog too!!!

Reply:

Regarding factorization of polynomials in F_p[x], there are some algorithms that improve the efficiency, but I do it by trial and error up to a cubic. This is because in these cases, a polynomial is reducible if and only if it has a root. It’s not so hard to plug in the numbers in small fields like F_7 to check for roots. That’s how I obtained the factor x+5.

In the third example, e refers to the identity element in the class group, that is, the class of the unit ideal O_K. In the computations of the previous paragraph, we showed that the set of \{e, [P_2], [P_3], [P_2P_3]\} really has order 4, and is also closed under the group law and taking inverses. For example, P_2^2=e.

Similar observations show that in fact, all elements have order 2. Thus, this set is a subgroup of Cl_K isomorphic to Z/2\times Z/2.
This is where 4\leq |Cl_K| comes from. But from the upper bound of 7, we had removed P_5, and hence, arrived at an upper bound of 6.

The group theory that follows then allows us to reach the conclusion. Because of that group theory, we can circumvent entirely a consideration of P_7.

As for principality, the main fact we use is that I is principal if and only if there is an element a\in I such that |N(a)|=N(I).

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