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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5
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4answers
88 views

What does $\Bbb{C}(X)$ refer to?

I have from a book (b) Let $E = \Bbb{C}(X)$. Then $\operatorname{Aut}(E / \Bbb{C})$ consists of the maps $X \mapsto \dfrac{aX + b}{cX + d}, ad-bc \neq 0\ldots$ Not sure what $\Bbb{C}(X)$ is. ...
1
vote
1answer
38 views

Basis of Q$(\sqrt[4]{3})$/Q

I want to show that Q$(\sqrt[4]{3})$/Q is algebraic. I am pretty sure that {1, $(\sqrt[4]{3})$} is a basis of this extension, and I was wondering if in general, for a (some irrational number) and the ...
0
votes
2answers
100 views

$\mathbb{Z}[x_{1},\dots,x_{n}]/I$ is a field therefore it's finite [duplicate]

I'd spent much time for this but didn't get any results.. Could u give me only the idea but not a full proof
0
votes
1answer
67 views

Why doesn't there exist a ring homomorphism between these two rings?

Why doesn't there exist a ring homomorphism between $\mathbf{Q}[x]/(x^2-2) $ and $\mathbf{Q}[x]/(x^2 -3) $? I see both rings are in fact fields as the polynomials are irreducible, further I know for ...
3
votes
0answers
50 views

Which of the following subsets of $\mathbb C$ is a field?

I'm not entirely sure that I understand the concept of fields fully so I'll give you the question and then I'll let you pick my brain and tell me if my logic is correct. Please note: I'm not just ...
0
votes
1answer
18 views

Notation of Rational Field Extensions

I was wondering what elements of the field $Q[\sqrt 2, \sqrt 3]$ look like? I think that $Q[\sqrt2 + \sqrt3]$ are elements of the form a + b ($\sqrt2 + \sqrt3$), where a and b are in Q. Are elements ...
0
votes
1answer
39 views

Field Extensions of Countable Sets

I was wondering, for a field F that is a countable set, and E/F algebraic, is field E necessarily countable as well? I know algebraic field extension implies every element of E is a root of a ...
6
votes
1answer
48 views

Extension fields isomorphic to fields of matrices

Suppose $K \subset L$ is a finite field extension of degree $m$. Is it true that there exists some natural $n$ such that $L$ is isomorphic to a subfield of $M^{n\times n}(K)$, the ring of $n\times n$ ...
0
votes
2answers
35 views

Elements of the Galois group of a polynomial acting as identity in the field $K$

For a field $K$, a polynomial $f \in K[X]$ and its splitting field $L$, we define the Galois group of the polynomial as $$\text{Gal}(f) := \text{Aut}(L/K)$$ The elements of $\text{Aut}(L/K)$ are the ...
2
votes
2answers
36 views

$L/K$ notation for field extensions

What does the notation $L/K$ for a field extension exactly specify? In group theory, such an object would be the group of cosets of a normal subgroup $K$ of a group $L$ and a similar usage exists for ...
1
vote
2answers
57 views

Theorem about embeddings of a field

Can you give me a reference* for the following theorem: Let $\mathbb{Q} \subset K\subset \mathbb{C}$ be an algebraic number field and $\alpha \in K$. If $\sigma(\alpha) = \alpha$ for all $\sigma ...
1
vote
2answers
62 views

Let $K$ be a finitely generated $F$-algebra. Show that $K$ a field. [duplicate]

Let $F$ be a field and let $K$ be a unital, associative, commutative, and entire $F$-algebra which, as a vector space, is finitely generated over $F$. Is $K$ a field? I know most of the ...
0
votes
0answers
14 views

Effect of Removing Element from Splitting Field

I know that $\mathbb{Q}({\sqrt[8]{2},\imath})/\mathbb{Q}(\sqrt{2}) \cong D_8$ where $D_8$ is the dihedral group of order 8. Does this imply that $\mathbb{Q}({\sqrt[4]{2}})/\mathbb{Q} \cong D_8$?
0
votes
1answer
33 views

Separable and irreducible polynomials over field with characteristic $p$

I am trying to show that $f(x)\in F[x]$ is irreducible and $char F=p$ then $f(x)=g(x^{p^e})$ for $g(x)$ irreducible and separable. I am working with the substitution map $\phi: F[x]\to F[x]$ which ...
1
vote
4answers
41 views

Complex roots of irreducible cubic in $\mathbb{Q}[x]$

Let $$f(x) = x^3 +ax^2 + cx + d \in \mathbb{Q}[x] $$ with one real root, and two complex roots: α and β α and β are conjugates. My task is to show that: $$β \notin \mathbb{Q}(α)$$ I'm confused as I ...
0
votes
1answer
37 views

Prove that the numbers of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational numbers, form a subfield of $\mathbb{C}$.

I'm having trouble proving that a multiplicative inverse exists in the following problem: Prove that the numbers of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational numbers, form a subfield of ...
0
votes
0answers
30 views

What examples of fields are known besides the basic ones?

I know the following fields $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ Other subfields of $\mathbb{C}$ Finite fields Algebraic closures of finite fields Fields of $p$-adic numbers Fields of ...
3
votes
1answer
59 views

The maximal unramified extension of a local field may not be complete

While reading my notes of a course in local class field theory, I arrived to a remark where it is said that given a complete discrete valuation field $K$, its maximal unramified extension $$K^{ur}= ...
0
votes
1answer
26 views

right rank(M) $\neq$ left rank(M)

In Artin, Galois Theory, we can prove that for any field $k$ the right column rank (noted RC, which is equal to the maximum number of independant columns with multiplication from right) is equal to ...
2
votes
1answer
44 views

When a multiplicative subgroup of a field generate a field?

Is it possible to find a field $F$ of prime characteristic which contains a non-trivial cyclic infinite subgroup $\langle x\rangle$ of $F^\times$ (the multiplicative group of $F$) such that the ...
2
votes
2answers
41 views

Square roots of $\Bbb F_p$

Can anyone please help me to show that $\Bbb F_{p^2}$ contains all the square roots of $\Bbb F_p$ where $p$ is a prime? Thanks for any help.
0
votes
0answers
37 views

If K $\subset$ L, K Galois over F, L is a splitting field of f(x) over F, can we say L Galois over F

Let $F$ be a field, $f(X) \in F[X]$ an irreducible polynomial of degree $n$ over $F$, $L$ a splitting field of $f(X)$ over $F$, and $\alpha \in L$ a root of $f(X)$. If $K$ is any Galois extension of ...
2
votes
1answer
49 views

Prove multiplication in fields is commutative

This is Problem $16$ from Halmos' Linear Algebra Problem Book. The problem asks whether or not multiplication must be commutative in a field. The solution uses the distributive properties ...
1
vote
1answer
49 views

Constructing an explicit tower with quadratic extensions.

I am so confused as to where to even start with this problem: a, b, c, and d are real numbers. Let α = a + bi, β = c + di, such that α = β2. Let Q be the set of rational numbers. Let K be a field, ...
1
vote
2answers
32 views

factor $x^3+2x^2+2x+1$ in $\mathbb{Z}_7[x]$

factor $x^3+2x^2+2x+1$ in $\mathbb{Z}_7[x]$ Is there a good strategy other than finding (guessing until you find a root) a root than using the division algorithm? Its not a very hard question to do ...
0
votes
0answers
30 views

False proof that $F_{3^2}$ contains $F_{3^4}$

Let's say we adjoin a second degree algebraic number to $F_3=\mathbb Z/3\mathbb Z$, it doesn't matter which. Then we get a field of $9$ elements, $F_{3^2}$. On the other hand, if we adjoin a fourth ...
0
votes
0answers
10 views

Identity with discriminant

Let $K$ be a field which is finite or of characteristic zero, let $L$ be an extension of finite degree $n$ of $K$, and let $σ_1,…σ_n$ be the $n$ distinct $K$-isomorphisms of $L$ into an algebraically ...
4
votes
3answers
177 views

``Minimal generating ring" for a field of fractions

In this answer and the linked MathOverflow post, it's shown that any field $F$ of characteristic zero contains a proper subring $A$ such that $F$ is the field of fractions of $A$. However, there is ...
0
votes
1answer
25 views

Dimension of certain field extensions of Q

My textbook says: Find dim$_\mathbb{Q}$ $\mathbb{Q}(\alpha, \beta)$, where: a. $\alpha^3 = 2$ and $\beta^2 = 2$. b. $\alpha^3 = 2$ and $\beta^2 = 3$. So by my calculations, $\alpha$ satisfies a ...
3
votes
2answers
55 views

Expand $(\vec{A}\times \nabla)\times \vec{B}$ using tensorial notation

I was given a task to prove $$(\vec{A}\times \nabla)\times \vec{B} = (\vec A \cdot \nabla)\vec B + \vec A \times \operatorname{rot} \vec B - \vec A \operatorname{div} B$$ using tensorial notation ...
0
votes
0answers
25 views

Degree of extension of a prime field $\mathbb Q$ to an algebraically closed field $\mathbb F$

Given an algebraically closed field $\mathbb F$ and its prime field $\mathbb Q$, what would be the degree of extension $[\mathbb F:\mathbb Q]$? Is it always infinite? It for sure is for $[\mathbb ...
0
votes
0answers
41 views

Prove or disprove the following:

Let K/F be algebraic. Then any $\sigma$ $\in$ Emb (K/F) induces an automorphism of K. (Emb(K/F) means all the embeddings of K $\rightarrow$ A (an algebraic closed field) over F. ) I have in ...
0
votes
1answer
29 views

Subfields of $F_2[x] / (x^3 + x + 1)$

What are the subfields of $F_2[x] / (x^3 + x + 1)$? I know $F_2$ is a subfield, and so is itself, but I'm not sure if there are any more. Thanks!
2
votes
1answer
44 views

Existence of Field with $p^n$ Elements.

If $p^n$ is a prime power, how can we show that there exists a field $F\supseteq\mathbb{F}_p$ such that $$x^{p^n}-x=(x-\theta_1)(x-\theta_2)\cdots (x-\theta_{p^n})$$ for some $\theta_i\in F$? Using ...
2
votes
0answers
33 views

Minimal polynomial of an algebraic element

We have to determine minimal polynomial over $\mathbb Q$ of element $1+2^\frac{1}{3}+4^\frac{1}{3}$. Here is my work, please tell me where is the mistake: We want polynomial with rational ...
2
votes
3answers
101 views

Field axioms: Why do we have $ 1 \neq 0$?

In the definitions of a field, we have $ 1 \neq 0$. I know that in regular multiplication $0 \times 1=0$ but for reciprocal we don't have inverse of $0$. But all the spaces and different definitions ...
1
vote
2answers
69 views

If $E,F$ are finite fields and $F\subseteq E,$ why is $E$ a finite-dimensional vector space over $F$?

I understand that if $E$ and $F$ are each finite and $E$ is a vector space over $F$, then $E$ must be a finite-dimensional vector space over $F$. However, my question is: why does $F\subseteq E$ imply ...
3
votes
1answer
49 views

Prove the polynomial is irreducible depending on the order of a field

I have to show that $f(x) = x^4+x^3+x^2+x+1$ is a irreducible polynomial in $F_p$ with $p \equiv 2 \pmod{5}$ or $p \equiv 3 \pmod{5}$. $f(x) \mid (x^5-1)$. This should be used for order of possible ...
2
votes
1answer
48 views

Necessary condition for the extension $F(\alpha) /F$ to be algebraic?

Let $K/F$ be an extension of fields and let $\alpha$ be an element of $K$ such that the set $\quad X = \{ \varphi(\alpha) \; | \; \varphi \in \operatorname{Aut}(K/F) \}$ is finite. Then, if $K/F$ ...
1
vote
1answer
41 views

The fundamental theorem of Galois theory

Let $E/Q$ be a Galois extension of degree $p^2$, where $p$ is a prime number. Prove that $L/Q$ is a Galois extension for any $L \in Intermediate(E/Q)$ and find $p$ if the cardinality of ...
2
votes
1answer
53 views

Find irreducible polynomial over $\mathbb F_9$

I am looking for a polynomial of degree $3$ in $\mathbb{F}_{9}$. How do I find one ? And if I have one how do I show that it is irreducible ? I would start with an irreducible polynomial in ...
0
votes
1answer
26 views

on a proof of the Primitive Element Theorem in zero characteristic

Here is a version of the Primitive Element Theorem and a proof in Fulton's Algebraic Curves: Question: My interpretation of the notation $(H,F)=(T-x) \in K'[T]$ is that the greatest common divisor ...
0
votes
1answer
105 views

Integral closure and field of fractions

I have a ring $R = \mathbb{Q}[t^2,t^5] \cong \frac{\mathbb{Q}[x,y]}{\langle x^5 - y^2 \rangle}$ (where the denominator is the ideal generated by $x^5 - y^2$). Now i have to compute the closure of $R$ ...
0
votes
2answers
72 views

Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
2
votes
1answer
53 views

Why is the algebraic closure of $\Bbb{Q}$ not a finitely generated $\Bbb{Q}$-module?

Let $\overline{\Bbb{Q}}$ be the algebraic closure of $\Bbb{Q}$. I am trying to show that $\overline{\Bbb{Q}}$ is not finitely generated as a $\Bbb{Q}$-module, however I do not know where to go with ...
1
vote
2answers
48 views

Can we find the generator of the Galois group of $x^{p-1}+x^{p-2}+…1$?

$p$ is a prime. We know that $x^{p-1}+x^{p-2}+...1$ is irreducible in Q[x]. And the splitting field of $x^{p-1}+x^{p-2}+...1$ over $Q[x]$ is $Q(\xi_p)$-the primitive pth root of unity. Now I want to ...
2
votes
1answer
40 views

Is there a field $K[\alpha]$ with $\alpha$ idempotent?

I have a field $K$ and an idempotent element $\alpha\not\in K$ (i.e., $\alpha^2=\alpha$), and I would like to form the ring $K[\alpha]$. Can this structure exist (1) in general and (2) with ...
1
vote
1answer
26 views

$F$ be a field of non-zero prime characteristic $p$ , is it true that there is only one group homomorphism $f:(F,+) \to (F$ \ $\{0\},.)$?

Suppose $F$ be a field of non-zero characteristic $p$ , is it true that there is only one group homomorphism $f:(F,+) \to (F$ \ $\{0\},.)$ ? I have tried taking $x \in F$ , then $px=0$ , so ...
2
votes
2answers
28 views

Showing that a given field is the splitting field of a given polynomial

Let $F = Z_2$; show that the splitting field of $f(x) = x^3 + x^2 + 1 \in F[x]$ is a finite field with $8$ elements. As $f$ has degree $3$, it is reducible if it has root in $F = Z_2$ but by ...
0
votes
1answer
21 views

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$.

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$. I was trying to show this. By the Primitive Element Theorem $\exists\alpha\in L$ such ...