New answers tagged

5 votes

Square root in number field

This answer is meant to answer only your second question. Claim. Let $K=\mathbb{Q}(\sqrt[3]{2})$ and $\alpha = \sqrt[3]{2}-\sqrt[3]{4} \in K$. Then there does not exist $\beta \in K$ such that $\beta^...
user avatar
  • 4,380
1 vote

Algebraic numbers which prescribed degree which does not belong to some fields

Proposition 2 is false, although perhaps only for $n = 4$ (and $t = 2$ or $t = 3$). If $t = 2$, every algebraic number $\gamma$ of degree $4$ is contained in $K_{4}$. Let $K$ be the Galois closure of $...
user avatar
  • 17.8k
5 votes

Algebraic numbers which prescribed degree which does not belong to some fields

Proposition 1 is true. Let $x$ be an element of degree $n$ whose Galois group is isomorphic to $S_n$. Let $F$ be a finite subset of algebraic elements of degree $<n$. I claim that $x$ is not in the ...
user avatar
  • 51.7k
11 votes

Sign and coefficients of fundamental unit of quadratic field

This might be useful: Stevenhagen, Peter, The number of real quadratic fields having units of negative norm, Exp. Math. 2, No. 2, 121-136 (1993). ZBL0792.11041. As Stevenhagen explains, if the ...
user avatar

Top 50 recent answers are included