# Tagged Questions

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### Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...
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### Field of all algebraic reals over $\mathbb{Q}$ has infinite order.

I am trying to show that field of all algebraic reals over $\mathbb{Q}$ has infinite degree. I guess that $$1,\sqrt{2},\sqrt[3]{2}, \sqrt[4]{2}, ...$$ are lineary independent but can't prove it.
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### The ring of integers of $\mathbf{Q}[i]$

Is there a relatively "simple" (in the sense that it does not require knowledge of algebraic number theory) proof that the ring of integers of the algebraic number field $\mathbf{Q}[i]$ is ...
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### Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
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### Suppose $γ$ is a $k$th root of unity that satisfies a quadratic equation $z^2−mz−n=0$ with $m,n\in\mathbb{Z}$. Then $k=3,4$ or $6$

Let $k\in\mathbb{Z}$ with $k>2$ and suppose $\gamma$ is a $k$th root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in\mathbb{Z}$.Then $k=3,4$ or $6$. My knowledge on ...
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### Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
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### $p > 2$ and ramification of archimedean places

Fix a rational prime $p$. I know that for a $p$-extension (ie. a Galois extension of degree a power of $p$) of an algebraic number field $k$, some places can not ramify: complex places cannot ...
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### Ideals in a real/complex number field?

Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$. Quick web search gave no ...
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### Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
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### When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let ...
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### What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
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### Division in a complete subring of a local field

Suppose $A$ is a complete subring of a local field such that a prime element $\pi$ belongs to $A$. Is it true that if $\beta=\pi^k u$ (with $k\ge 0$ and $v(u)=0$) and $\beta\in A$ then also $u\in A$? ...
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### Field norm defined with or without absolute value?

I'm studying about the valuation for Euclidean Domains and quadratic fields $\mathbb{Q}(\sqrt{\theta})=\{ \alpha: \alpha=a+b\sqrt{\theta}, a,b \in \mathbb{Q} \}$ and I'm not sure whether we start with ...
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### Eigenvalues of Multiplication in algebraic number field

Finally I have a new question which is puzzling me. I am currently reading a manuscript so there is no reference or link I can provide. I will try to put all the information needed in here. This ...
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### Number fields containing the values of a character

This is a basic question- Let $K$ be an algebraic number field, $\Gamma$ a finite group, and $R(\Gamma)$ the ring of virtual characters of $\Gamma$ with values in the algebraic closure $\mathbb{Q}^c$ ...
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### Ideal norm in a quadratic field

Let $K=\mathbb{Q}[\sqrt{d}]$ be a quadratic field with discriminant $d_K$, let $\mathfrak{a}=(a,\frac{b-\sqrt{d_K}}{2})$ be an ideal. Does the norm $N(\mathfrak{a})=a$? How to prove it?
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### Element of with n0n-zero trace

Let $F$ be a field of characteristic $p$ and $K$ a finite, separable extension of $F$ such that $p \mid [K : F]$. I want to show that there must exist an element of $K$ with non-zero trace. One idea ...
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### Extensions of number fields

Let $L|K$ be an extension of number fields and consider the corresponding (integral) extension of ring of integers: $R_L|R_K$. Note that $R_L$ and $R_K$ are finitely generated over $\mathbb{Z}$, hence ...
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### what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
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### Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
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### Nonreal units in totally imaginary number fields

Suppose we are given a totally imaginary number field $L$ of degree greater $2$, is it possible that all units of $L$ lie in $\mathbb R$? This is true for almost all quadratic imaginary number fields, ...
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### Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
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### Show the two fields are not isomorphic

Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$. ...
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### Roots of unity in $\mathbb{Q}(\zeta_p)$ for $p$ an odd prime

I am confused about a last step in a proof that the only roots of unity in $\mathbb{Q}(\zeta_p)$ are $\pm\zeta_p^j$, where $\zeta_p=e^{2 \pi i/p}$, $p$ is an odd prime and $1\leq j\leq p-1$. So far ...
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### Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.

I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
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### The composite of all unramified extensions inside an algebraic closure

I'm reading Ch.II, $\S$ 7 of Neukirch's Algebraic Number Theory and I'd be really grateful if someone could help me understand the following: Let $K$ be a complete valued field wrt a non-archimedean ...
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### Galois group of CM fields

I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois ...
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### Unramification and compositum

The background is: a field $K$ complete with respect to a discrete valuation $|\ |$. We write $A$ and $k$ for his discret valuation ring and the residue field of $A$. We assume that $K$ and $k$ are ...
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### A sum of products symmetric in the images under all the embeddings

Let $\mathbb{Q}\subset K\subset \mathbb{C}$ where $K$ is a finite extension of $\mathbb{Q}$. Let $\sigma_1, \dots, \sigma_n$ be all the embeddings $K\rightarrow \mathbb{C}$. Is it true that elements ...
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### Show that a number field is isomorphic to a quotient $\mathbb Q[x]/(f)$

Let $K$ be a number field of degree 3. Show that $K$ is isomorphic to a quotient $\mathbb Q[x]/(f)$, with $f = x^3 + ax + b$ in $\mathbb Z[x]$ irreducible in $\mathbb Q[x]$ (without using the result ...
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### An element is integral iff its minimal polynomial has integral coefficients.

This is from Algebraic Number Theory by Neukirch Let $A$ be an integral domain which is integrally closed, K its field of fractions, $L|K$ a finite field extension, and $B$ the ...
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### Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
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### Trace and Norm of a separable extension.

If $L | K$ is a separable extension and $\sigma : L \rightarrow \bar K$ varies over the different $K$-embeddings of $L$ into an algebraic closure $\bar K$ of $K$, then how to prove that ...
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### Why is this isomorphism $M \otimes_K L \stackrel{\simeq}{\longrightarrow} M^{[L:K]}$ an isomorphism of $M$ - algebras?

Suppose that $L/K$ is a finite separable extension of fields and let $M$ denote the Galois closure of $L$. Let $\textrm{Hom}_K(L,M)$ denote the set of all $K$ - algebra homomorphisms from $L$ to $M$. ...
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### Motivation on traces, norms and discriminants

I am looking for some motivation for the definitions of trace, norm and discriminant (in the context of finite field extensions). For example (but not limited to) any interesting theorems proved using ...
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### “Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
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### Cyclotomic extensions of $\Bbb Q$

Let $n>4$, and $(h,n) = 1$. How to show that $[\mathbb{Q}(\tan 2 \pi h/n):\mathbb{Q}]= \phi(n)$ or $\phi(n)/2$ or $\phi(n)/4$ respectively if $\gcd(n,8)<4$ or $\gcd(n,8)=4$ or ...
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### Equivalence of Valuations - Trouble Understanding Proof

I want to complete the proof of the following theorem. Here is what I have got so far: Theorem Every non-euclidean valuation $v$ on a number field $K$ is equivalent to $v_{\mathfrak p}$ for some ...
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### “Place” vs. “Prime” in a number field.

I have been trying to make sense of what a "place" is. In the setting of a number field, is a place simply a prime ideal? My understanding is that one can complete a number field with respect to a ...
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### How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
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### Positivity of the norm of an element of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be an $l$-th primitive root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field and $\alpha$ be a non-zero element of ...
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### Finding the matrix of multiplication by $\theta^2$, where $\theta^3 - 3\theta + 1 = 0$

This is a problem from a on-line source which yet comes with a solution (self-studier; not h.w.). Let $E = \mathbb Q(\theta)$, where $\theta$ is a root of the irreducible polynomial \[ X^3 -3X + 1. ...
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### Analogy between trace pairing on a number field and the dot product.

How is the trace pairing function $(x,y) \mapsto Tr(xy)$ on a number field an analogue of the dot product in euclidean space? (This is a view shared by Keith Conrad and can be found in his notes ...
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### The level of a $p$-adic number field

First I define the level of field. The level of a field $\mathbb K$ is the least $n$ such that $−1$ is a sum of $n$ squares in field, and is denoted by $S(\mathbb K)$. I know that the level of ...
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### What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?

Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
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### What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And ...
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### $\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ ...
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### A Number Field that's Galois over $\mathbb{Q}$ is either totally real or totally imaginary

I came across the following assertion in the Wikipedia article about totally imaginary number fields. Let $K/\mathbb{Q}$ be an algebraic number field that is Galois over $\mathbb{Q}$. Then ...
### Prove that $\mathbb{Q} \left( \sqrt[n]{p}\right) \neq \mathbb{Q} \left( \sqrt[n]{q}\right)$.
Given two distinct prime numbers $p$ and $q$, how can we prove that $\mathbb{Q} \left( \sqrt[n]{p}\right) \neq \mathbb{Q} \left( \sqrt[n]{q}\right)$ where $\sqrt[n]{p}$,$\sqrt[n]{q}\in \mathbb{R}$ and ...
Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then ...