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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3
votes
1answer
17 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?
-1
votes
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?
2
votes
1answer
36 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 ...
0
votes
0answers
31 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
vote
1answer
23 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
votes
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
vote
2answers
47 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
vote
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
votes
1answer
37 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
votes
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
votes
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 ...
0
votes
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.
0
votes
1answer
23 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
votes
1answer
55 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
472 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
27 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
36 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
vote
1answer
19 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
32 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
60 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
votes
0answers
11 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
votes
0answers
40 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
votes
0answers
77 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
vote
2answers
34 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
vote
1answer
32 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
votes
2answers
18 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
vote
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
votes
2answers
327 views
0
votes
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 ...
-3
votes
0answers
16 views

To prove Degree of extension is less than equal to 2 [closed]

Let K be finite field extension of F and K is equal to its algebraic closure. Then $[K:F] \leq 2$
2
votes
1answer
45 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 ...
0
votes
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 ...
1
vote
2answers
41 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) ...
0
votes
4answers
84 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. ...
0
votes
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 ...
-2
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!
1
vote
0answers
17 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
vote
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 ...
1
vote
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
vote
1answer
49 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 ...
1
vote
1answer
36 views
3
votes
1answer
643 views

Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
1
vote
1answer
29 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 ...
3
votes
3answers
415 views

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 “$\subseteq$” ...
0
votes
0answers
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 ...
2
votes
3answers
32 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 ...
8
votes
2answers
442 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
vote
2answers
21 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 ...
0
votes
0answers
18 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 ...