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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1answer
32 views

Why this field extension has such basis?

Here is an argument for finding a basis for $\mathbb{Q}$. Since $X^2-2$ is irreducible in $\mathbb{Q}[X]$, $\{1,\sqrt{2}\}$ is a $\mathbb{Q}$-basis for $\mathbb{Q}(\sqrt{2})$. Since $X^4-10X^2+1$ ...
-1
votes
3answers
44 views

Let $K\subset \mathbb{R}$ be a field. Prove that $\mathbb{Q}\subset K$

I don't have any ideas for this one. The hint the book gives is to prove that $\mathbb{Z}\subset K$ but I don't see how that helps.
4
votes
1answer
35 views

On diagonizability of commutating matrices

Let $A$ and $B$ be $n\times n$ matrices over the Galois Field of order $p$ ($p$ is a prime). Suppose that $A$ and $B$ are diagonizable matrices and that they commutate. Is it possible to make them ...
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vote
3answers
35 views

How to guarantee existence of a finite field [duplicate]

Existence of a finite field: Solution: I can understand that if I have a finite field $F$ of characteristic $p$ where $p$ is prime then I can consider $\mathbb Z_p$ as its prime field and hence $F$ ...
1
vote
1answer
37 views

Isomorphism of Quotient Fields

Are the two fields $\mathbb R[x]/\langle(x^2+1)\rangle $ and $\mathbb R[x]/\langle(x^2+x+1)\rangle $ isomorphic? Solution Now $\mathbb C$ is algebraic over $\mathbb R$ .Consider the evaluation ...
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votes
1answer
37 views

Artin's Algebra Book Problem:Prove that splitting fields of $x^3+ex+6$ over $Q(e)$ and $x^3+\pi x+6$ over $Q(\pi)$ are isomorphic.

Assume that $\pi$ and $e$ are transcendental. Let $K$ be the splitting field of $f(x)=x^{3} + \pi x + 6$ over $F = Q(\pi)$ (a) Prove that $[K : F] = 6$. (b) Prove that $K$ is isomorphic to the ...
1
vote
1answer
55 views

How to do multiplication in $GF(2^8)$?

I am taking an Internet Security Class and we received some practice problems and answers, but I do not know how to do these problems , an explanation would be greatly appreciated Try to compute the ...
1
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1answer
75 views

Extension of $\mathbb{C}$

I want to prove that $\mathbb{C} \subset \mathbb{k}$ and $\dim_{\mathbb{C}}\mathbb{k} \leq \aleph_{0} \implies \mathbb{C = k}$($\mathbb{k}$ - field). If $\mathbb{k} \ni e, e^2,\dots,e^n$ are ...
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0answers
22 views

Is algebraic closure of rational functions field Puiseaux series?

Consider a field of rational functions over algebraicly closed field. Is its algebraic closure isomorphic to Puiseaux series over the field?
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votes
1answer
29 views

Prove the fractional field $Q(\mathbb{R})$ of the integral domain $\Bbb R$ is $\mathbb{R}$.

Prove the fractional field $Q(\mathbb{R})$ of the integral domain $\Bbb R$ is $\mathbb{R}$. I proved Prove the fractional field $Q(\mathbb{Z})$ of the integral domain $\Bbb Z$ is $\mathbb{Q}$. Using ...
1
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1answer
36 views

Does it follow from field axioms that the sum of multiplicative identity with its additive inverse is the additive identity? [closed]

In other words, is it true (and if so why) that 1-1=0? (1 and 0 being multiplicative and additive identity elements)? Some background. I'm reading Mathematical Background: Foundations of ...
0
votes
1answer
51 views

Why is $F(x,y)$ isomorphic to $F(x)(y)$?

Here $F$ is a field, $F(x,y)$ is the field of rational functions in $x$ and $y$ with coefficients in $F$, and $F(x)(y)$ is the field of rational functions in $y$ with coefficients in $F(x)$, the field ...
0
votes
3answers
68 views

Is $M_{2}(\mathbb{R})$ an algebra over $\mathbb{C}$

Is $M_{2}(\mathbb{R})$ an algebra over $\mathbb{C}$? I tried the homomorphism $\phi(a+bi)=\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$. The problem is $\phi$ is homomorphism but not into the ...
0
votes
0answers
40 views

Prove the fractional field $Q(\mathbb{Z})$ of the integral domain $\Bbb Z$ is $\mathbb{Q}$.

prove $Q(\mathbb{Z})= \mathbb{Q}$ I have already proven that for some integral domain R, if we take $R \times R \backslash \{0\}$ it is a field under the operations. $(a,b)+(c,d) = (ad+bc, bd)$ and ...
1
vote
1answer
41 views

If $p=4k+3$ then $\mathbb{Z}_{p}[\sqrt{-1}]$ is a field

I want to know about an elementary proof (at the level of elementary number theory) of the fact If $p$ is a prime number of the form $4k+3$ then $\mathbb{Z}_{p}[\sqrt{-1}]$ is a field where $$ ...
1
vote
1answer
49 views

BIG Intermediate fields in an infinite tower

This question arose out of the answer by Brandon Carter to the following question: Infinite dimensional intermediate subfields of an algebraic extension of an algebraic number field Define an ...
0
votes
2answers
41 views

Infinite dimensional intermediate subfields of an algebraic extension of an algebraic number field

Let $K$ be an algebraic number field, i.e. a finite extension of $\mathbb{Q}$. Let $L/K$ be an infinite dimensional algebraic extension of $K$. Are there infinitely many infinite dimensional ...
0
votes
1answer
29 views

show that $K=\Bbb F_{q^m}$, where m is the order of $q$ in the group of units $\Bbb {Z}_n^*$ of the ring $\Bbb Z_n$.

Let $q$ be power of a prime $p$, and let $n$ be a positive integer not divisible by $p$. We let $\Bbb F_q$ be the unique upto isomorphism finite field of $q$ elements. If $K$ is the splitting field of ...
0
votes
1answer
32 views

How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$.

I have two questions: 1) How to deduce the irreducible factors of $x^4 +1$ in $\Bbb F_3$. Clearly the irreducible factors will be of degree $2$. But can anyone calculate it for me? 2) I have proved ...
0
votes
1answer
24 views

Algebraic Indepence of Functions over Infinite Field

Can someone point in the right direction to a reference or give me an idea of the proof of the following fact. My field theory is rusty: Let $K$ be an infinite field of arbitrary characteristic. ...
0
votes
1answer
48 views

Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
2
votes
3answers
45 views

Frobenius injective for finite fields - what about $\mathbb{F_{p^n}}$

Quick question about the Frobenius endomorphism. My lecture notes and wikipedia say that the Frobenius is injective for finite fields. However, if we look at $\mathbb{F_4}$, we have $$\text{Frob}(2) = ...
5
votes
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
42 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
102 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
59 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
63 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 ...
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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
60 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
27 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
38 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
50 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 ...
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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, ...
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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 ...
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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 ...