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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2
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1answer
82 views

Decomposition of Normal Field Extensions

I am trying to prove the following homework problem: Let $K/k$ be a normal field extension and $K_i$ and $K_s$ be intermediate extensions so that $K_i/k$ and $K/K_s$ are purely inseparable and ...
1
vote
1answer
44 views

Showing that $(\mathbb{F}_q[x]/(f_i(x)))^{F_q} = \mathbb{F}_q$

In a proof in my syllabus of a number theory course, they use that $$(\mathbb{F}_q[x]/(f_i(x)))^{F_q} = \mathbb{F}_q$$ where $f_i(x)$ is irreducible, $F_q$ is the frobenius automorphism of ...
7
votes
1answer
937 views

Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in ...
8
votes
3answers
2k views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
0
votes
1answer
134 views

Computing the generators of a real cubic field extension of $\mathbb{Q}$

What's a good example where computing the generators of a real cubic field extension of $\mathbb{Q}$ is nontrivial? I usually see these fields specified in terms of generators, is there a good ...
6
votes
2answers
349 views

Field Isomorphisms

Suppose $F/L$, $F'/L$, $L/K$ finite extensions of fields. If $F$, $F'$ isomorphic over $K$ then does it follow that they are isomorphic over $L$? I think probably not, but I can't come up with a ...
6
votes
3answers
720 views

On a formula of the norm of an element of a finite extension of a field

Theorem Let $F$ be a field. Let $K$ be a finite extension of $F$. Let $[K : F]_i$ be the inseparable degree of $K/F$. Let $\bar{K}$ be an algebraic closure of $K$. Let $S$ be the set of $F$-embeddings ...
6
votes
3answers
250 views

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?
1
vote
1answer
99 views

Finding a minimal polynomial in char $2$.

[Some (useless) context: the following problem comes from a problem in Algebraic Geometry, where I have to show that a certain morphism $\textbf P^2\to \textbf A^2$ is inseparable of degree $2$.] Let ...
2
votes
1answer
284 views

General Primitive Element Theorem

I have a proof of the primitive element theorem for subfields $K$ of $\mathbb{C}$ which relies on there being infinitely many elements in $K$. In a book I've been reading it states that every finite ...
20
votes
2answers
667 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
6
votes
2answers
487 views

Why aren't there any coproducts in the category of $\bf{Fields}$?

Well the question is stated in the title. I dont know much about field theory and i was suprised when i read it on wikipedia please provide some examples thanks in advance
1
vote
0answers
131 views

Abel-Ruffini Theorem Clarification

Let $n \geq 5$. I want to show that $\exists p\in \mathbb{Q}[X]$ of degree $n$ with roots which are not possible to find via radicals and rational functions from the coefficients of $p$. I've got the ...
4
votes
2answers
178 views

Find all finite fields k whose subfields form a chain: if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$. So, I understand that I'm trying to ...
2
votes
2answers
207 views

Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reading texts about field extensions I've come across the following two definitions for the simple extension $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ ...
8
votes
1answer
2k views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge ...
5
votes
1answer
271 views

Finite Field Extensions and the Sum of the Elements in Proper Subextensions (Follow-Up Question)

I recently posted the following question, to which this question is a follow-up. Regardless, my question here will be self-contained. Let $F$ be a finite field, and let $u,v$ be algebraic over $F$. ...
5
votes
2answers
132 views

Finite Field Extensions and the Sum of the Elements in Proper Subextensions

Let $F$ be a finite field, and let $u,v$ be algebraic over $F$. Consider the fields $F(u,v),F(u)$ and $F(v)$. Must it be the case that $F(u,v) = F(u)$ or $F(u,v) = F(u+v)$?
1
vote
1answer
212 views

Isomorphic group to the multiplicative group of a field.

Suppose that I have an abelian group $G$ and an epimorphism from $G$ to $(K^*)\times (K^*), $ where $K$ is an algebraically closed field. Is it true that $G$ cannot be isomorpic to $K^*.$?
4
votes
5answers
400 views

Write down a $\mathbb Q$-Basis for $\mathbb Q(\alpha, i)$ and show that $\mathbb Q(\alpha, i) = \mathbb Q(\alpha + i)$.

I have found a $\mathbb Q$-basis for $\mathbb Q(\alpha)$, where $\alpha$ is a root of $x^{3}-x+1$, to be $\{1, \alpha, \alpha^{2}\}$ & a $\mathbb Q(\alpha)$-basis for $\mathbb Q(\alpha, i)$ to be ...
9
votes
1answer
220 views

Question about a property certain algebraic extensions $E/K$ (not necessarily separable) have.

A few days ago I found this question here on math.stackexchange, which gave a sufficient criterion for a separable, algebraic extension $E/K$ to be an algebraic closure of $K$. However it was claimed ...
10
votes
3answers
1k views

Galois Group of $(x^3-5)(x^2-3)$

I am having some trouble calculating the Galois group (over $\mathbb{Q}$) of $(x^3-5)(x^2-3)$. I can see the splitting field is ...
0
votes
1answer
100 views

The size of Galois group

In continuation to my last post: In class we saw an example that says: $n=[\mathbb{F}_{p^n}:\mathbb{F}_{p}]=|\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p})|$ ; ($p$ is prime). My thoughts are ...
3
votes
2answers
415 views

Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
1
vote
1answer
231 views

Why is the cyclotomic polynomial over $\mathbb{Q}$?

As defined in Wikipedia (and this is the same definition I was given in class), it is not clear to me why the cyclotomic polynomial is over $\mathbb{Q}$. It is over $\mathbb{C}$, but I don't see a ...
8
votes
1answer
524 views

Why $E$ is the algebraic closure of $K$?

Let $E/K$ be a separable, algebraic extension such that every noncostant polynomial in $K[x]$ has a root in $E$, then $E$ is an algebraic closure of $K$. Could you help me to solve this exercise? ...
7
votes
1answer
252 views

Algebraic Extensions and Separability

I have been wrestling with the following problem for the past few hours, but I have made no progress whatsoever, namely: Let $K/k$ be an algebraic extension with characteristic $p>0$ and let ...
6
votes
2answers
307 views

Why is this extension of Galois?

Let $F$ be a subextension of $\mathbb{C}$ maximal with respect to not containing $\sqrt2$. Let $K/F$ be a finite extension with $K\subset\mathbb{C}$. Then $K/F$ is of Galois and $[K:F]$ is a power of ...
3
votes
1answer
759 views

Definition of Separability Degree

For an assignment, I am trying to determine the separability degree of some algebraic field extension $L/K$. The definition of the separability degree of polynomial is not difficult to find at all, ...
5
votes
1answer
131 views

Example of field $K$ with $\mathrm{char}(K) > 0 $, such that $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$

I'd like to fine an example of field $K$ and elements $\alpha, \beta$ such that $\mathrm{char}(K) = p> 0 $, $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$. This obviously can't ...
1
vote
1answer
44 views

Why $H\neq N_G(H)$?

Let $K$ be a field, $f(x)$ a separable irreducible polynomial in $K[x]$. Let $E$ be the splitting field of $f(x)$ over $K$. Let $\alpha,\beta$ be distinct roots of $f(x)$. Suppose ...
4
votes
3answers
1k views

No extension to complex numbers?

Complex numbers are 2D. It is a commonly sited result that there is no 3D or 4D analogue of the complex numbers. I just want to be clear on exactly what this result says: It is impossible to ...
9
votes
1answer
1k views

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 ...
3
votes
3answers
149 views

Why is $p(x)=x^2+t\in\mathbb{F}_2(t)[x]$ irreducible?

Why is $p(x)=x^2+t\in\mathbb{F}_2(t)[x]$ irreducible? The only argument I can think of is that $\sqrt t$ doesn't seem like a rational function, but I tried to prove this and got stuck when I had to ...
6
votes
1answer
413 views

Does $\varphi(1)=1$ if $\varphi$ is a field homomorphism?

Is it by definition that $\varphi(1)=1$ if $\varphi$ is a field homomorphism ? My field theory lecture said that yes, but now $\varphi\equiv0$ is not a field homomorphism... What is the 'standard' ...
0
votes
1answer
101 views

why this extension doesn't contain a subextension of degree 2?

Consider the polynomial $f(x)=x^4+6x^3+32x^2+17x-15$ and let $\alpha\in\mathbb{C}$ be a root of $f$. How can I show that $\mathbb{Q}(\alpha)$ has no subfield of degree 2 over $\mathbb{Q}$? I have an ...
3
votes
1answer
73 views

If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$

If I understand correctly, Arturo Magidin says in this comment that the following is true. If $E\subset F$ is a finite field extension, then $|\operatorname{Aut}(F/E)|\leq (F:E).$ I understand ...
3
votes
1answer
175 views

Associativity of norms in inseparable extensions

Let $K$ be a field. Let $L/K$ and $E/L$ be finite extensions. Let $α$ be an element of E. Let $N_{E/K}(α)$ be the norm of $α$, i.e. the determinant of the regular representaion matrix of $α$. It is ...
3
votes
1answer
292 views

What is the Galois group of the extension $\mathbb F_3(x^4)\subset\mathbb F_{3^2}(x)?$

Square brackets $[\;]$ will denote taking the ring of polynomials, and round brackets $(\;)$ will denote taking the field of rational functions. My homework assignment from about a month ago had the ...
4
votes
2answers
319 views

When a field extension $E\subset F$ has degree $n$, can I find the degree of the extension $E(x)\subset F(x)?$

This is not a problem I've found stated anywhere, so I'm not sure how much generality I should assume. I will try to ask my question in such a way that answers on different levels of generality could ...
6
votes
1answer
1k views

Sums and products of algebraic numbers

How does one go about proving that the sums and products of two algebraic numbers over a field $F$ (say $a,b\in K$, where $K/F$ is a field extension) is also algebraic? If we call $f_a$ and $f_b$ ...
1
vote
1answer
314 views

In what vector space does Dedekind's lemma live?

Dedekind's lemma in field theory says this: Let $E$ and $L$ be fields, and $\sigma_1,\ldots,\sigma_n:E\longrightarrow L$ be distinct field homomorphisms. Then $\sigma_1,\ldots,\sigma_n$ are ...
5
votes
2answers
246 views

Need help determining the Galois group of an extension

In an assignment I got I have been asked to try to determine when $E/F$ is a Galois Extension, and determine the Galois group of such an extension. $\textbf{Context:}$ $F$ is any field of ...
12
votes
3answers
8k views

what is difference between a ring and a field

The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra ...
2
votes
1answer
133 views

Field homomorphism $\varphi:\mathbb{Q}(\sqrt[3]{2})\to\mathbb{K}$ where $\mathbb{K}$ is the splitting field of $x^3-2$

Denote $\mathbb{K}$ as the splitting field of $x^3-2$ . I wish to find $\Gamma:=\left\{ \varphi:\mathbb{Q}(\sqrt[3]{2})\to\mathbb{K}|\forall q\in\mathbb{Q}:\varphi(q)=q\right\} $. Sind $\varphi$ is ...
1
vote
1answer
363 views

Separable polynomials - a question about a paragraph from “Abstract Algebra” by Dummit and Foote

I have a question about the following paragraph from Dummit and Foote on separable polynomials: We now investigate further the structure of inseparable irreducible polynomials over fields of ...
3
votes
2answers
480 views

Surjectivity of the Frobenius map

I wish to prove the following claim: Let $\mathbb{F}$ be a finite field of characteristic $p$ then the map $a\to a^p$ is surjective Dummit and Foote's Abstract Algebra says that this map is ...
2
votes
2answers
69 views

Why all the roots of $x^n -1$ are multiple over $\mathbb{F}_p$ if $p|n$?

This is a claim in the book "Abstract algebra" in one of the examples, can someone explain this please ? I know that the derivative of this polynomial is identically $0$ so there is a multiple root, ...
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vote
2answers
1k views

When the group of automorphisms of an extension of fields acts transitively

Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ ...
3
votes
2answers
1k views

Minimal polynomial of a twelth root of unity

I am attempting to find the minimal polynomial for $\omega=\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}+i\frac{1}{2}$ over $\mathbb{Q}$. I'm doing this in the context of cyclotomic ...