# Questions tagged [field-extensions]

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### Identity for compositum and intersection of fields

Let $k$ be an arbitrary base field and $K, L, M$ some fields over $k$ contained in a fixed overfield $\Omega$. Question: Are there some "reasonable" assumptions (ie beyond a bunch of really ...
70 views

### Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$

Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
1 vote
99 views

### Degrees of trigonometric numbers

For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers. What is its degree? That is, what is the minimal degree of ...
1 vote
46 views

### Galois action on blow-ups related to field extensions of infinite degree

Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...
284 views

### How do I extend the $2$-adic absolute value to prove Monsky's Theorem?

In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$... 168 views

### Existence of intermediate field extensions for tamely ramified p-adic extensions

Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
98 views

### Any connection between extension of algebraic structure and forcing of set theory?

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
1 vote
160 views

### Fields with restrictions on their finite extensions: Given $n\in\mathbb{N}_{>1}$ which fields $F$ do not have extensions of degree $n$?

$\DeclareMathOperator\char{char}$This question is inspired by the MSE question Example of a non-algebraically closed field without quadratic extensions. To repeat: Given $n\in\mathbb{N}_{>1}$ ...
291 views

### Is the minimal polynomial of an algebraic formal Laurent series always separable?

Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$ always separable? An example of non separable polynomial comes from ...
155 views

### A dimension problem related to an abelian simple extension of a field

$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
465 views

### Absolutely irreducible representation and splitting field

Let $A$ be a finite-dimensional algebra over a field $F$. A representation $M$ of $A$ is called absolutely irreducible if $M\otimes_FE$ is irreducible as a representation of $A\otimes_FE$ for all ...
1 vote
83 views

### Indeterminacy locus of an algebraic function

Let $K=\mathbb{C}(t_1,\dots,t_n)$ be the field of rational functions, $f$ an algebraic function over $K$ and assume the field extension $K(f)/K$ is non-solvable. Is it possible to characterise the ...
114 views

### Finding elements in a real extension of $\mathbb{Q}$ that are close to some number in $\mathbb{R}$

Let's consider a set of numbers that one knows to high precision, and one knows or has a strong suspicion that `exact versions of these numbers' (see below) belong to a real extension of $\mathbb{Q}$. ...
279 views

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### The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

I asked this question over a year ago on Math.StackExchange but I didn't get an answer. In his famous treatise On spirals, Archimedean used a spiral to square the circle and trisect an angle. There ...
1 vote
368 views

### A quantity associated to a field extension

Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space. A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...
1 vote
89 views

### Algorithms for Polynomials Over a Real Algebraic Number Field, a reference

I need to find "Algorithms for Polynomials Over a Real Algebraic Number Field Ph.D. thesis, University of Wisconsin, Madison (1974) by Rubald". However I cannot find it online nor in my ...
1k views

### Counter example of a radical extension that is not Galois/normal over $\mathbb{Q}(\omega)$?

Most proofs of Galois theorem stating that "an equation is solvable in radicals if and only if its Galois group is solvable," show the left to right direction by induction on the height of ...
1 vote
189 views

### Reference book for Galois extension [closed]

I need a reference for field extension and Galois extension (like an introduction) please. Thank you.
1 vote
291 views

### A composition of a simple extension and a separable extension is simple

Let $K/L/M$ be a tower of finite field extensions with $K/L$ separable and $L/M$ simple (in the sense of being generated by a single element). How does one show that $K/M$ is also simple? I know that ...
580 views

### The variety induced by an extension of a field

If $K$ is a finitely generated field extension of $k$, then there exists an irreducible affine $k$-variety with function field $K$. The idea is that if $x_1, \dots, x_n$ are generators of $K$ under $k$...
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### Modal Principles of Field Extensions

In 2007 (with more work done later), J. Hamkins and B. Löwe found that the ZFC provably valid principles of forcing are the assertions of S4.2. In the introduction, they mention field extension as a ...
459 views

### Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$

I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows: As a set $D=E\oplus uE \oplus u^2 E$ where $u$...
187 views

### Extension of isomorphism of fields

I know this is not a "research Mathematical question", but this is a question that I would ask to my Math colleagues, but no one could give me a readily "yes-no" answer. Unfortunately, this is not a ...
1k views

### What are the primes that are ramified?

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we ...