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 ...

learn more… | top users | synonyms

3
votes
3answers
116 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
1
vote
1answer
35 views

A field of characteristic zero is perfect

How do you prove that a field F of characteristic zero is perfect, or rather that every irreducible f(x) in F[x] is separable? Thank you!
0
votes
0answers
14 views

intersection is isomorphic to $Aut(F)$

Prove that for any finite field $F$ there exist two subgroups of $GL(|F|,\mathbb{R})$ whose intersection is isomorphic to $Aut(F)$
2
votes
1answer
25 views

Field automorphisms and fixed fields

I currently have to cope with field automorphisms. I already understood that any field automorphism of $\mathbb{C}$ must fix all elements in $\mathbb{Q}$. My question is the following: Assume a ...
5
votes
2answers
106 views

Generators of the Relations of a Galois Extension

Let $K$ be a Galois extension of $\mathbb{Q}$ of degree $n$. Pick some primitive element and take the roots $a_1, ..., a_n$ of its minimal polynomial. Then the evaluation map $\mathbb{Q}[x_1, ..., ...
5
votes
1answer
197 views

Sigma-Algebra: Is it an Algebra, Field, or Something Else?

The Wikipedia page for $\sigma$-algebra says this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on measure theory, where the topic of sigma-algebra ...
1
vote
2answers
55 views

Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$

Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is ...
3
votes
2answers
179 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
1
vote
1answer
56 views

Extending Homomorphism into Algebraically Closed Field

If we are given a homomorphism $g$ between a field $k$ and an algebraically closed field $\Omega$, and a field $k'$ which is a finite algebraic extension of $k$, how do we extend $g$ to a homomorphism ...
4
votes
1answer
38 views

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$. Attempt: $F \subset F(a^3) \subseteq ...
2
votes
1answer
58 views

A field extension of prime degree

Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$ Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. ...
1
vote
2answers
28 views

existence of some algebraic closure

I was curious somethings when I studied algebraic closure in Hungerford's algebra. I know that every field has an algebraic closure. But I have the following question. Let K be a field and F an ...
1
vote
2answers
48 views

What are subfields of $\mathbb{C}$?

I only took the first undergraduate abstract algebra course, so i don't know (at all) what Galois theory is about. I'm asking this question since i'm not sure of the definition of inner product space ...
2
votes
1answer
32 views

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f(x)$ and $\deg g(x)$ are relatively prime.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\gcd(~\deg g(x),\deg f(x)~)=1$. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$ ...
3
votes
1answer
30 views

Field theorem: impossible to satisfy three equations simultaneously in an integer field?

Suppose $a_0,a_1,b,c,d$ with $a_0 \neq a_1$ are integers, or elements of any ${\mathbb Z}_p$ for prime $p$. A while ago I was attempting to find a set of such numbers that satisfied the equations ...
3
votes
1answer
52 views

Question about calculating the degree of a finite field extension.

I have a question about calculating the degree of a finite field extension over $\mathbb{Q}$. This is problem 18 in chapter 1 of Patrick Morandi's Field and Galois theory. The problem asked to show ...
0
votes
0answers
40 views

Why do nth roots (radicals) have closed forms whilst other polynomial roots do not?

The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class ...
7
votes
1answer
78 views

Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
0
votes
0answers
34 views

Extending a morphism to a finite algebraic field extension [duplicate]

I am trying to understand the proof of Theorem 5.21 in Introduction to Commutative Algebra, and am stuck on the portion underlined in red (note that $$\Sigma := \{(A,f) \mid A \text{ is a subring of ...
2
votes
0answers
56 views

Question on a finite field extension of $\mathbb{Q}$

I have a polynomial $p(x) \in \mathbb{Q}[x]$ and is irreducible over $\mathbb{Q}$. Let it be of degree $n$ and $\alpha_1, ..., \alpha_n$ be its roots. I know that $$ \mathbb{Q}(\alpha_i) \cong ...
4
votes
2answers
46 views

Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
0
votes
0answers
7 views

Solutions to a polynomial equation in a PAC field not lying in a subfield

Suppose $f(x,y)$ is an absolutely irreducible polynomial over a PAC (pseudo algebraically closed) field $K$ such that $x,y$ actually appear in $f$. Let $L$ be a proper subfield of $K$. Are we ...
3
votes
3answers
96 views

Finitely many embeddings of a finite extension in an algebraic closure

So I'm reading through Lang's Algebra, and he keeps saying something along the following lines: "Let $K$ be a finite extension of a field $k$ and let $\sigma_1,\ldots,\sigma_r$ be the distinct ...
3
votes
0answers
49 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field ...
0
votes
0answers
35 views

A question regarding a lemma in Perrin's Algebraic Geometry.

Algebraic Geometry by Perrin says the following: Let $k$ be an uncountable algebraically closed field and let $K$ be an extension of $k$ whose dimension is at most countable. Then $K=k$. He ...
0
votes
1answer
40 views

Is the field formed by algebraic elements of an extension field over $F$ isomorphic to $F[t]$?

Say $K/F$ is a field extension. The elements in $K$ that are algebraic over $F$ form a subfield of $K$. Is this subfield isomorphic to $F[t]$? What would this isomorphism look like? This is not a ...
1
vote
1answer
53 views

Prove that any two bases of a field extension have the same cardinality.

Suppose $E$ and $F$ are subfields of $\mathbb{R}$ with $F\subseteq E$. Prove that any two bases of $E/F$ have the same cardinality. The definition of a basis I am using is any finite set $S\subseteq ...
1
vote
2answers
53 views

There always exists a subfield of $\mathbb C$ which is a splitting field for $f(x)$ $\in$ $Q[X]$?

So I've been studying field theory on my own, and I just started learning about splitting fields. Based on my understandings if a polynomial, $f(x)$ $\in$ $Q[X]$, then there should be always a ...
2
votes
2answers
36 views

Transitivity Property of Separable Extensions

I was looking for some proof for the transitivity property of separable field extensions. Although this might sound like a very well-known fact and is referred to frequently, I do not seem to find a ...
1
vote
3answers
37 views

Quadratic number fields containing primitive roots of unity

A problem from Michael Artin's Algebra (Second Edition) from Fields: Determine the quadratic number fields $\mathbb{Q}[\sqrt{d}]$ that contain a primitive $n$th root of unity, for some integer $n$. ...
2
votes
2answers
75 views

How do elements of $\mathbb{R}(xy,x+y)$ look like?

I have problems with determining what are typical elements of such field $\mathbb{R}(xy,x+y)$ In one indeterminate it is easier as $\mathbb{R}(x)=\Bigl\{\frac{f(x)}{g(x)}, g(x)\neq 0, ...
1
vote
1answer
55 views

Describe the elements in $Q(\pi)$

Describe the elements in $Q(\pi)$ Attempt: $Q(\pi)$ is the smallest field which contains $Q$ and $\pi$ We know that $\nexists~ f(x) \in Q[x]$ such that $f(\pi)=0$ Hence, $Q[x]/\langle p(x) ...
0
votes
0answers
16 views

Is there a direct way of proving that all splitting fields are isomorphic?

Given $f(x)\in F[X]$, let $E,E'$ be two extension fields of $f$ over $F$, then $E \approx E'$. Now, I've seen a proof involving directly constructing an isomorphism, but I'm searching for another ...
1
vote
1answer
36 views

Every field with characterisitic $p$ contains the field $\mathbb{Z}_p$

I seem to hold a very loose grasp of the concept of fields - I've encountered this question: Show that every finite field with characteristic $p$ contains $\mathbb{Z}_p$ (i.e. $\mathbb{Z}_p = ...
0
votes
2answers
30 views

Zeroes of f(x) in a splitting field $E $ have the same multiplicity

Let $f(x)$ be an irreducible polynomial over a field $F$ and let $E$ be a splitting field of $f(x)$ over $F$. Then all the zeroes of $f(x)$ in $E$ have the same multiplicity. The proof of this ...
1
vote
1answer
59 views

Are order isomorphic real closed fields isomorphic?

There are counterexamples to order isomorphisms of ordered fields being field isomorphisms, see Is the multiplicative structure of a totally ordered field unique?. However, Wikipedia suggests that for ...
0
votes
2answers
38 views

Discrete valuations of the rational numbers

I'm trying to find every discrete valuation on the field of rational numbers. If $a\in \mathbb Q$, we can write $a=p^j\frac{x}{y}$, where $p$ is a prime number and $p\nmid x$ and $p\nmid y$. We can ...
2
votes
1answer
37 views

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$.

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$. I have some queries regarding this theorem of existence of ...
2
votes
3answers
153 views

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...
2
votes
1answer
42 views

Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
2
votes
4answers
151 views

Is there a ring $K$ such that $\mathbb R\subsetneqq K\subsetneqq \mathbb C$?

Is there a ring $K$ such that $\mathbb R\subsetneqq K\subsetneqq \mathbb C$? and if $K$ is a local ring, is there such a ring? I need this result be negative to prove a question I'm working. I need ...
2
votes
2answers
43 views

For which values of $a\in\mathbb ℤ/3\mathbb ℤ$ is the quotient $\mathbb ℤ/3\mathbb ℤ[x]/(x^3+x^2+ax+1)$ a field?

I'm trying to solve the following problem: Determine for which values of $a\in\mathbb{Z}/3\mathbb{Z}$ the quotient $Q_a=(\mathbb{Z}/3\mathbb{Z})[x]/(x^3+x^2+ax+1)$ is a field. I see two options: ...
2
votes
1answer
33 views

Isomorphic algebraic closures.

I've just stared learning the Galois Theory so my question might be trivial, but could someone give me an example of two different algebraic closures of the same field? Cause I don't get how they can ...
0
votes
2answers
69 views

Prove that $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{-1}$ with a,b,c∈Q is a number of the form $d+e\sqrt[3]{2}+f\sqrt[3]{4}$ with $d,e,f∈Q$ [duplicate]

Prove that $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{-1}$ with $a,b,c∈Q$ is a number of the form $d+e\sqrt[3]{2}+f\sqrt[3]{4}$ with $d,e,f \in Q$ I'd like to do this without using too much fancy ...
1
vote
1answer
50 views

Confusion regarding proof of a proposition in Field Theory (Dumb Question)

Question is to prove that : For finite extensions $E/k$ and $F/k$ Prove that $[EF:k]\leq [E:k][F:k]$ where $EF$ is the smallest field extension which contains both $E$ and $F$ Supposing ...
3
votes
1answer
40 views

Field of Definition of an Algebraic Group

Linear Algebraic Groups- James E. Humphreys Chapter-XII Let $K$ be an algebraically closed field and $k$ be a arbitrary sub-field of $K.$ A closed set X in $A^n=K\times ...$(n times)$\times K$ is ...
2
votes
1answer
24 views

Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
13
votes
1answer
267 views

Families of Polynomials Irreducible in $\mathbb{Z}$ but reducible in $\mathbb{Z}/p\mathbb{Z}$ for all primes $p$.

I am wondering if there exist classification of polynomials that are irreducible in $ \mathbb{Z}$ but reducible $\pmod p$ for all primes $p$. I am aware that $\Phi_n$ has this property if ...
0
votes
1answer
82 views

Query on an example of Morandi's Field and Galois Theory, regarding the degree of a field extension.

I am going through Morandi's Field and Galois Theory, and I am looking at Example $1.5$, Chapter I. It says, (more or less) If $k$ is a field, let $K=k(t)$ be the field of rational functions in ...
1
vote
3answers
45 views

Extensions of degree $1$.

My doubt is very simple: Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$? I'm trying to prove the equality without success. Thanks in advance