# Tag Info

• 137k
Accepted

### How Dirichlet proved Dirichlet's unit theorem for general number fields?

Dirichlet's proof is described in Number Theory: Algebraic Numbers and Functions (starting on page 48). Dirichlet did not use Minkowskiâ€™s theorem; he proved the unit theorem in 1846 while ...
• 178k

### In which cyclic cubic number fields does there exist this type of unit?

The answer to question 1 is yes. Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio w=\frac{(x_1-1)(x_2-x_3)}{(...
• 2,391
Accepted

### Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

More is true. Let $K/F$ be a field extension. Let $X$ be the vanishing set of some polynomials in $K[X_1,\dots, X_n]$. If $X$ contains a Zariski dense set of points with coordinates in $F$, then $X$ ...
• 137k
Accepted

### Are the rationals definable in any number field?

According to R. S. Rumely, Undecidability and Definability for the theory of global fields, AMS Trans., 262, pp. 195-217, prime subfield is always definable in global field, and in number case, you ...
• 4,446

### 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 ...
• 36.2k
Accepted

### irreducibility of the polynomial $x^4 +1$

Among $-1$, $2$, and $-2$, there are $0$, $1$ or $3$ squares. This determines the irreducible factorization of $X^4+1$. If none is a square, it is irreducible; If only $-1=i^2$ is a square, the ...
• 60.4k

### Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Asking in terms of $B$ how many $a_n$ are needed is equivalent to asking the following question: What is the largest $N$ such that there exists two quadratic characters $\chi_1, \chi_2$ of ...
• 137k
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### Upper bounds for regulators of real quadratic fields

Stephane Louboutin has several papers on getting explicit bounds for $L(1,\chi)$, for $\chi$ a character $\pmod q$. They're all of the strength of $1/2 \log q +$ an explicit constant. Some of his ...
• 43.3k

### Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

I don't want to toot my own horn, but I coauthored a paper on this topic with Marvin Jones. One direction of our result is conditional on the Birch and Swinnerton-Dyer conjecture (see the first remark ...
• 20k
Accepted

### Irreducibility of polynomials over some number fields

Lemma. Let $K$ be any number field, and $p$ a prime unramified in $K$. Then $X^n-p$ is irreducible over $K$. Proof. It suffices to show that the field $L = K(\sqrt[n\ \ ]{p})$ has degree $n$ over $K$. ...
Accepted

### class number of prime degree field with prime conductor

Maybe I am making a mistake here, but let me try: Let $H$ be the Hilbert class field of $K$. Then $H\cap \mathbb{Q}(\zeta_p)=K$ as otherwise one prime in there should be totally ramified and ...
• 8,341
Accepted

### Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

Barrucand and Cohn (MR0249396, Note on primes of type $x^2+32y^2$, class number, and residuacity. J. Reine Angew. Math. 238, 1969, 67--70) have proved that for primes $p \equiv 1 \textrm{ mod } 8$, ...
• 20.5k
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### What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$

There is an existential definition of $\mathbb{Z}$ in the rational function field $\mathbb{R}(t)$ by a beautiful result of Denef using elliptic curves (Proposition 2 of The diophantine problem for ...
• 2,041
Accepted

• 2,041

### Galois embedding question for dihedral groups

The answer is "no", in general, since there may be local obstructions. Suppose, for example, that $k$ and $n$ are odd prime powers, and let $L/\mathbb{Q}$ be the unique intermediate ...
• 12.8k
Accepted

### Standard conjecture on u-invariants?

For the classical $u$-invariant of fields of characteristic $\neq 2$, some known results are: The $u$-invariant of formally real fields is $\infty$. If $K$ is an algebraically closed field, its $u$-...
• 1,422
The following article discusses p-adic expansions in $\overline{\mathbb{Q}}_p$ and of $\mathbb{C}_p$. Algebraic $p$-adic expansions, David Lampert, Journal of Number Theory 23 (1986), 279–284.