Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

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1answer
33 views

A problem regarding field theory: determine the fixed field

I came across this problem in N. Jacobson's Basic Algebra (Vol I): Let $E = (\mathbb{Z}/(p))(t)$, where $t$ is trasncendental over $\mathbb{Z}/(p)$. Let $G$ be the group of automorphisms generated ...
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1answer
27 views

Curious about field extensions and the relationship between the automorphisms of the fields' Galois groups

Let's say we have field extensions $F(\alpha)$, $F(\beta)$ and $F(\alpha, \beta)$of a field $F$. My question is this: if $\sigma \in {\rm Gal}(F(\alpha)/F)$, is it correct to say that $\sigma \in {\rm ...
1
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0answers
23 views

elements of a finite field vs its characteristic

What is the difference saying, '' a finite field with $q$ elements'' or ''a field with characteristic $p$'' ? a finite field must be of prime characteristic and vice versa, what is the difference ...
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2answers
22 views

Example of Galois extension over Q which is not cyclotomic

So, prof. introduced Galois extensions yesterday and I do apologise if I did not get something correctly. So, if I am right every finite extension of finite field is almost obviously Galois(using ...
0
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1answer
19 views

Splitting field and automorphisms

I know that if $K$ is a field and $f\in K[x]$, then there exists a splitting field of $f$ on $K$. If one has two isomorphic fields $K_1$ and $K_2$ (say $\sigma$ an isomorphism) and $f\in K_1[x]$, ...
2
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1answer
58 views

Is the primitive element of an extension of a finite field expressible as a linear combination of adjoined elements?

Suppose $F$ is a field, and $\alpha,\beta$ are algebraic, separable elements over $F$. If $F$ is infinite, the proof of the primitive element theorem gives some $c\in F$ such that ...
4
votes
1answer
33 views

Discriminant of a trinomial $x^n+ax^m+b$

I am trying to compute the discriminant of the trinomial $x^n+ax^m+b$. I have tried using resultants but cannot see how to approach it. Any hints?
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0answers
33 views

Galois Group is $\mathbb{Z}/4\mathbb{Z}$.

Let $K \subseteq L$ be a Galois field extension with Gal$(L/K) \cong \mathbb{Z}/4\mathbb{Z}$. Show that $L$ is the splitting field of a polynomial $f(x)=(x^2 −a)^2 −b$ for elements $a,b \in K$ such ...
1
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1answer
39 views

Galois Extension.

Suppose K is a finite field extension of $\mathbb{Q}$. Let K ⊆ L be a Galois field extension and K ⊆ K′ be a finite field extension. Show that K′ ⊆ K′L is a Galois field extension and $$\text{Gal}(K′L ...
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0answers
42 views

How many elements are in the field of fractions $\Bbb Z_3(t)$?

As in exercise for my Galois Theory course I am supposed to find the number of elements in the field of fractions $\Bbb Z_3(t)$. I am very confused as to how to approach this question because I ...
0
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1answer
38 views

Will two different subgroups of a Galois group have different fixed fields?

I'm trying to figure out will two different subgroups of a Galois group have different fixed fields. Intuitively, I think they have the same fixed fields. But I am not sure. Anyone has ideas?
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0answers
21 views

Galois Group of Splitting Field, $S_4$

I've shown that the polynomial $x^4+px+p \in \mathbb{Q}[x]$, where $p$ is prime, is irreducible by Eisenstein's criterion. However, it remains to be shown that the Galois group of the splitting field ...
1
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1answer
27 views

Intermediate field extensions of $\mathbb{Q}(\zeta_3)/\mathbb{Q}$

I'm trying to determine the intermediate field extensions of $\mathbb{Q}(\zeta_3)/\mathbb{Q}$. The minimal polynomial of $\zeta_3$ is $x^3+1$, which has roots $\zeta_3, \zeta_3^2$ and $-1$. Therefore, ...
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1answer
27 views

Finite fields and generators of Galois group [closed]

I'm trying to find the order and all generators of the Galois group of $GF(256)$ over $GF(16)$? How to approach this problem?
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0answers
38 views

Non-existence of a particular type of tower of number fields

I have number fields $\mathbb{Q}\subset K\subset H$ where $K\subset H$ is Galois. I want to show that is is impossible for a rational prime $p\in\mathbb{Z}$ to remain first inert in $K$ but then for ...
1
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1answer
30 views

Does $K/E$ and $E/F$ being normal mean $K/F$ is normal?

Let $F\subset E \subset K$ be fields. Suppose that $K/E$ and $E/F$ are normal. Is $K/F$ also normal? I feel that this statement is not true in general but I cannot find a counter-example. Any ...
3
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1answer
12 views

Separable polynomial with splitting field an unramified extension?

I am trying to prove a theorem and it seems that I need that an irreducible polynomial $f(x)$ that is separable over $\mathfrak{p}$ has its splitting field an unramified extension of ...
1
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2answers
54 views

Find $Gal(K/\mathbb{Q})$ and show that $K/\mathbb{Q}$ is normal where $K=\mathbb{Q}(a)$

Let $K=\mathbb{Q}(a)$ and $a$ is a root of $x^3+x^2-2x-1 \in \mathbb{Q}[x]$. Find $Gal(K/\mathbb{Q})$ and prove that $K/\mathbb{Q}$ is normal. I just noticed that $a^2-2$ is also a root of the ...
1
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0answers
16 views

Solution space of semilinear equation

I found the following lemma and the corollary in a paper and I don't know how to prove them. Therefore I was wondering if one of you could help me. Let $E$ be a field, $ \sigma: E \rightarrow E$ ...
5
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1answer
43 views

Showing normalizer of Galois group

Let $E/F$ be a Galois extension, and let $B$ be an intermediate field between $E$ and $F$. Let $H$ be the subgroup of $Gal(E/F)$ that maps $B$ into itself (but does not necessarily fix $B$). Prove ...
5
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2answers
48 views

Finding $p(x)$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q[x]}/\langle p(x)\rangle$

I am trying to find a polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q[x]}/\langle p(x)\rangle$. This is what I tried to do: Consider ...
2
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0answers
29 views

What came first: pythagoras number or pythagorean fields? [migrated]

Which concept was first introduced: the pythagoras number of a field or pythagorean fields? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the ...
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1answer
30 views
0
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1answer
26 views

Proof Enquiry, Field of order $p^n$ [duplicate]

I want to prove that there exists an inclusion $\mathbb{F}_{p^a} \hookrightarrow \mathbb{F}_{p^b}$ iff $a \vert b$. Suppose that $a \vert b$, then $b =ac$ for some $c \in \mathbb{Z}$. Consider then ...
3
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1answer
56 views

Unramified algebraic extensions of local fields

This is a basic question from Neukirch's Algebraic Number Theory, Prop. 7.2: Fix a non-Archimedean local field $K$. Let $L/K$ and $K'/K$ be two extensions inside an algebraic closure $\bar{K}/K$ and ...
11
votes
5answers
478 views

Fields of arbitrary cardinality

So I took an introductory abstract algebra course a few semesters ago, and we were shown that groups and rings can both be made into products, i.e. if I have some group $G$ (resp. ring $R$) and some ...
2
votes
1answer
30 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
0
votes
0answers
52 views

Which polynomials are resultants?

Let $f(x,y),g(x,y)\in\mathbb{Q}[x,y]$ with degrees $\deg(f)=m,\deg(g)=n$. Considering these polynomials as univariate polynomials in $y$ over the field $\mathbb{Q}[x]$, the resultant ...
1
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1answer
21 views

Calculating the Matrix of a Transformation Using Bases of Field Extensions

I'm trying to understand this topic in my Abstract Algebra class: Suppose that we have a finite field extension $L/F$ and let us choose $a \in L$. We'll define the transformation $T_{a} : L \to L$ ...
2
votes
1answer
34 views

Field Trace/Norm and Matrix Trace/Norm (Dummit and Foote 14.2.31(c)).

I can't quite figure out this final part to 14.2.31 in Dummit and Foote, 3rd edition. I'm given $K/F$ is a finite field extension of degree $n$, and $\alpha\in K$. I've shown that the map ...
0
votes
2answers
61 views

Find the Galois group of $x^3-5$ over $\mathbb{Q}$.

In this case, the roots of $x^3-5$ are $\{\sqrt[3]{5},\omega\sqrt[3]{5},\omega^2\sqrt[3]{5}\}.$ I think $\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5})$ is the splitting field of $x^3-5.$ Then, ...
0
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0answers
12 views

Finding Primitive Elements of Separable Function Field Extensions

Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields ...
2
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0answers
41 views

Field extensions that decompose into towers of degree$\leq n$ extensions

Let $F$ be a field and let $n$ be a natural number. Consider the class of field extensions $E/F$ that decompose into towers $E=E_k/E_{k-1}/\cdots/E_1/E_0=F$ such that $[E_{i+1}:E_i]\leq n$ for ...
10
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0answers
87 views

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
1
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2answers
35 views

Ring homomorphism of polynomial ring

Let $R\left [ x \right ]$ be a Polynomial ring. Let R be a ring If $R\left [ x \right ]\rightarrow R$ $f\left [ x \right ] \mapsto f\left ( 0 \right )$ is a ring homomorphism ...
1
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1answer
33 views

Degree of a finite field extension

Let $i,\sqrt{3}\in\mathbb{C}$. I know that both are algebraic over $\mathbb{Q}$. Hence $[\mathbb{Q}(i\sqrt{3}):\mathbb{Q}]=\deg(i\sqrt{3},\mathbb{Q})$. This is equal to 2 since ...
0
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2answers
20 views

If $v$ is algebraic over $K(u)$, for some $u\in F$, and $v $ is transcendental over $K$, then $u$ is algebraic over $K(v)$

If $v$ is algebraic over $K(u)$ for some $u\in F$, $F$ is an extension over $K$, and $v $ is transcendental over $K$, then $u$ is algebraic over $K(v)$. I came across this problem in the book Algebra ...
0
votes
2answers
31 views

Showing a subset $K$ is a subfield of a field

Let $F$ be a field and let $K$ be a subset of $F$ with at least two elements. Prove that $K$ is a subfield of $F$ IF, for any $a,b$ ($b\neq 0$) in $K$, $a-b$ and $a\cdot b^{-1}$ belongs to $K$. ...
1
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0answers
12 views

unique ringhomomorphism from the Field of fractions to another field

$R$ is a ring, $L$ a field and $K$ the fraction field constructed from $R$. For any injective ring homomorphism $f=R \rightarrow L$, there is a unique ring homomorphism $\tilde{f}:K \rightarrow L$ ...
3
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2answers
330 views
0
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0answers
19 views

Finding a condition

So, we have $Q(\sqrt{2},\sqrt{3})=Q(\sqrt{2}+\sqrt{3})$. $\supset$ is absolutely trivial and $\subset$ holds because $\frac{1}{\sqrt{3}+\sqrt{2}} = \sqrt{3}-\sqrt{2} \in Q(\sqrt{2},\sqrt{3})$ and ...
2
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1answer
50 views

Field theory: an equality involving the number of homomorphisms from an extension $E$ of $F$ to $\overline{F}$

First some notation. Let $F$ be a field, $E$ an algebraic extension of $F$ and $\overline{F}$ the algebraic closure of $F$. Let $\{E:F\}$ represents the number of non-zero homomorphisms from $E$ to ...
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0answers
27 views

Explicit matrix representation of an algebraic extension

This may be considered an extension of this question. Let $\mathbb{F}$ be a field, and let $p(X)\in\mathbb{F}[X]$ be an irreducible polynomial. Let $\mathbb{F}_p$ be the extension of $\mathbb{F}$ by ...
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2answers
42 views

Showing automorphisms on $\mathbb{C}(x)$

Let $\mathbb{C}(x)$ denote the field of rational functions over $\mathbb{C}$, the field of complex numbers. Consider the six mappings $\phi : \mathbb{C}(x) → \mathbb{C}(x)$ defined by $\phi_{1}:f(x) ...
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4answers
86 views

Show that the field of real numbers has an infinite proper subfield but no finite subfields.

Show that the field of real numbers has an infinite proper subfield but no finite subfields. $\mathbb{Q}$ is an infinite subfield and as $|\mathbb{Q}| < |\mathbb{R}|$, it is also a proper subfield. ...
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2answers
53 views

Why is $\mathbb{Q}(\sqrt{2}\sqrt[3]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$

Why is $\mathbb{Q}(\sqrt{2}\sqrt[2]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$ "obvious"? My book states this as obvious, but then proves the opposite inclusion. I would have thought that ...
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1answer
32 views

Field extension if element fixed by only identity [closed]

Suppose that $E$ is a Galois extension of $F$ and that $α \in E$ is left fixed by only the identity in $\text{Gal}(E/F)$. Prove that $E = F (α)$. Please suggest how I should proceed. Thanks!
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0answers
18 views

Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
1
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1answer
38 views

Elements in a Field of size $27$

I constructed the Field $$F_3[x]/<1 + 2x + x^3>$$ as the question asked to construct a field of size $27$ and I understood everything up to this point. The solution then says the elements in ...
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0answers
11 views

On the question of the Galois group of some polynomial. [duplicate]

I want to ask you some question on the Galois group of some polynomial. Let $p_1<p_2<\cdots<p_n$ be distinct prime numbers. Let's consider $f(t)=(t^2-p_1)(t^2-p_2)\cdots(t^2-p_n)\in ...