Brandon Carter
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 Jul27 answered Reference request: principalization theorem Jul23 comment When does a Galois group of a quintic have order divisible by three? Jyrki, I meant the Frobenius density theorem (it is referred to as the "theorem of Frobenius" in the second paragraph of the section I linked to). It certainly follows from Chebotarev, as Chebotarev generalizes Frobenius' theorem, but the full power of Chebotarev is not necessary and the proof of Frobenius density is much simpler, see mathoverflow.net/questions/136025/frobenius-density-theorem. Jul22 answered When does a Galois group of a quintic have order divisible by three? Jul21 revised Remark 4.23.4 in Hartshorne. edited tags Jul21 comment Why is the degree of this field extension $[K(x,y): K(x^p, y^p)]= p^2$? Hint: $K(x^p, y^p)$ is the compositum of $K(x^p,y)$ and $K(x,y^p)$. Jul2 comment About Mordell's Theorem (Elliptic Curves) The proof can be simplified some where the base field is $\mathbf{Q}$ and $n = 2$, even to the point of only needing to understand some basic algebraic number theory (e.g. the possible discriminants of quadratic fields). Even in that case, I don't see where the ideal class group or Dirichlet's unit theorem play a role. Jul2 comment About Mordell's Theorem (Elliptic Curves) @sdf: Where do you think that Dirichlet's unit theorem comes into this? The usual proof of the finiteness of the Selmer group comes from showing that the $n$ Selmer group lies inside of the set of continuous homomorphisms from the absolute Galois group to $E[n]$ that are unramified outside of a finite number of places. But then you can use that such a homomorphism factors through the Galois group of the maximal abelian extension of exponent $n$ which is unramified outside of those specified places. This extension is finite by CFT. Jul2 comment About Mordell's Theorem (Elliptic Curves) J.S. Milne's notes are a great resource for both local and global class field theory. Jul2 answered About Mordell's Theorem (Elliptic Curves) Jul2 revised Using an ellipse, do all inscribed angles have to be congruent? edited tags Jun23 answered Some questions on elliptic curves over finite fields Jun23 answered moduli of the conjugates of a cyclotomic integer Jun23 comment Every finite abelian extension of Q contains a totally real subfield of index 2? The statement from Wikipedia says "An abelian extension of $\mathbf{Q}$ is either totally real, or contains a totally real subfield over which it has degree two." This means that the totally real subfield is index 2, not degree 2, as @Zev mentioned. This is otherwise false as stated; for example, the compositum of a cyclic cubic extension of $\mathbf{Q}$ and an imaginary quadratic field will not satisfy the incorrect condition. Jun9 reviewed Reject Solving the equation $f^{-1}(x)=f(x)$ Jun3 comment Determine $[K(\zeta_{16}):K]$ when $K=\mathbb{F}_7, \mathbb{F}_9, \mathbb{F}_{17}$ You have all of the computations backwards. For example, $x^8 + 1$ splits completely over $\mathbf{F}_{17}$. You should be looking for the order of $p$ mod $16$, rather than the other way around. May25 comment Order of prime ideals over split primes in the class group. @James: You also have extra units if $d = -3$. But my point was to get you to see that the possibility that $a = 0$ exists. My example above is not unique to real quadratic fields; it works just as well with looking at $5$ in $\mathbf{Q}(\sqrt{-5})$. May25 comment Order of prime ideals over split primes in the class group. @James: Why do you think that $a + b \sqrt{d} \sim a - b \sqrt{d}$ implies that $b = 0$? What is the unique (ramified) prime lying above $3$ in $\mathbf{Q}(\sqrt{3})$, for example? May24 comment Is there a classification of ideals of $\mathcal O_K$ ($K$ quadratic) over ramified and split primes depending on $d \pmod 4$? You should look up Dedekind's criterion for splitting of primes, particularly as it relates to quadratic fields. This will tell you how to determine which primes are split, and to find generators for the primes lying over $p$. May20 revised How to find inverse of generator of a finite field? edited tags May9 comment Determining whether a given algebraic number is an algebraic integer The question was not to find a basis for $\mathcal{O}_K$, but to check whether a particular element is integral.