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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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$ ...
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

Prove that a commutative ring without proper ideals is a field [duplicate]

Let $R$ is a commutative ring which has no proper ideals. Prove that $R$ is a field.
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1answer
25 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 ...
2
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1answer
29 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 ...
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0answers
51 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 ...
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1answer
20 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 $T_{a} : L \to L$ ...
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 ...
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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 ...
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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 ...
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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 $K$. ...
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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 ...
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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, ...
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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 ...
2
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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 ...
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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 $K$. ...
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2answers
19 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 ...
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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 $\tilde{f}:K \rightarrow L$ ...
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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
49 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 ...
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 ...
5
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1answer
40 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 ...
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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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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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votes
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
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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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 $\Omega$ ...
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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 ...
1
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1answer
39 views

Why is $\mathbb{F}_5[x]$ a Jacobson ring? [closed]

As the question title suggests, why is $\mathbb{F}_5[x]$ a Jacobson ring?
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54 views

What does the irreducible polynomial over $\mathbb{Q}$ and $\mathbb{Q}(i)$ mean?

So here'as the problem Find the monic irreducible polynomial $g(x) \in \mathbb{Q}(x)$ for $i+ \sqrt{3}$ over $\mathbb{Q}(i)$ Erm...huh? Okay, so a minimal polynomial $m$ is what I need, and ...
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1answer
30 views

Algebraic function fields

I am trying to understand what an algebraic function field is, so i was looking for some examples. The example on Wiki says: Given a polynomial ring $k[X,Y]$. Consider the ideal generated by the ...
2
votes
3answers
35 views

Why number of bases of $\mathbb{F}_p^2$ equals order of $GL_2(\mathbb{F}_p)$?

Artin, Algebra, Chapter 3, Ex. 4.4 I can prove (b), viz., that The order of $GL_2(\mathbb{F}_p)=p(p+1)(p-1)^2$ The order of $SL_2(\mathbb{F}_p)=p(p+1)(p-1)$ However, I have no idea how to prove ...
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0answers
19 views

Field of the form $\{a+bi|a,b\in \mathbb{F}_p\}$

Artin, Algebra, Chapter 3, Ex 1.11 Consider whether the set of symbols $\{a+bi|a,b\in\mathbb{F}_p\}$ forms a field, if the laws of composition are made to mimic addition and multiplication of complex ...
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2answers
22 views

The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
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0answers
40 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
8
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2answers
445 views

What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
1
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0answers
24 views

Minimal polynomial of a primitive element for Galois extensions with Galois group $S_n$

Let $K$ be a global field, $f(x)\in K[x]$ be an irreducible separable polynomial and $L$ be the splitting field of $f(x)$. Suppose that the Galois group of $L$ over $K$ is the symmetric group ...
1
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1answer
52 views

Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is ...
4
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1answer
36 views

Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, ...
2
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1answer
46 views

About transitive subgroups of symmetric group $S_n$

When I am studying Galois theory I came across some problems: Let $S_n $ be the symmetric group on $n$ letters($|S_n|=n!$).How to determine all the transitive group $G$ of $S_n $ ( A subgroup $G$ ...
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3answers
45 views

rational numbers field axioms

Let $\mathbb Q$ be the rational number field. Is the group $K=\left\{\left.\begin{pmatrix} a & 2b\\ b & a \end{pmatrix}~\right|~ a,b\in \mathbb Q\right\}$ a field with the regular addition ...
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3answers
24 views

Finding the fixed subfield (Galois theory)

Let's say we are working with the field extension $\mathbb{Q}(\gamma)$, where $\gamma$ is the seventh root of unity. I know my basis for this extension will thus be: $\{1, \gamma, \gamma^2, ...
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1answer
41 views

By “shifting” , what does this mean?

I am looking at the solutions to a problem that asks me to show The only subfields of $\mathbb{Q}(i,\sqrt{5})$ are ...
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1answer
40 views

Some natural question on subfield of Galois extension

Let $\alpha,\beta\in \mathbb{\overline{Q}}$ and assume $\deg(\text{Irr}(\alpha,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q(\alpha)})=2$. Then I strongly guess that ...
0
votes
1answer
43 views

If $f(a)=f(a+1)$, then $F$ has characteristic $0$.

Suppose $f\in F[x]$ is irreducible, $E$ is the splitting field of $f$, and for some $a\in E$ we have $f(a)=f(a+1)=0$. Then $F$ has characteristic $0$. I'm not sure how to use the last assumption: ...
0
votes
1answer
14 views

Finding the fixed subfield corresponding to a cyclic subgroup of the Galois group

Let's say I have a field extension $E$ of some field $F$ and I also know the Galois group of $E$ over $F$. Suppose I have a subset of this Galois group which is cyclic, thus generated by some ...
1
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1answer
46 views

Degree of the field extension

I need to determine the degree of the field extension $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3}}))/\mathbb{Q}$. I've determined that the minimal polynomial of $\sqrt{(2+\sqrt{2})(3+\sqrt{3}})$ is ...
1
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1answer
24 views

Prove that $F \subset \sigma L$ is also radical.

Suppose that we have finite extensions $F \subset L \subset M$ and $\sigma \in Gal(M/F)$ and assume that $F \subset L$ is radical. Prove that $F \subset \sigma L$ is also radical. Since the ...
7
votes
1answer
44 views

Unramified primes of splitting field

I would like to show the following: Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ ...