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
8 views

Abelianization of the absolute group and maximal abelian extension

Let $K$ be any field, $\overline K$ is the separable closure of $K$ and $K^{ab}$ is the maximal abelian extension of $K$. I want to prove the following relation $$G(\overline ...
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2answers
17 views

Direct Sum algebraic number

I'm doing a course in algebraic number theory and I don't understand the direct sum notation. How can the direct sum of fields be equal to a field? I thought the direct sum is a Cartesian product, ...
2
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1answer
33 views

Galois group isomorphic to $\mathbb Z$

Does exist an example of a Galois extension $L/K$ such that $\text{Gal}(L/K)\cong \mathbb Z$? Thank you.
3
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2answers
32 views

Invertible matrices modulo $29$. [duplicate]

Consider the $n\times n$ matrices with elements in $\mathbb{Z}_{29}$. How many of these are invertible? In total there are $29^{n^2}$ matrices of of dimension $n\times n$. Now I need to find how ...
4
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0answers
20 views

Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear ...
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0answers
17 views

What are all the Algebraic Elements of $F(t)$ over $F$?

Let $F$ be a field and $t$ be a variable. Let $F(t)$ be the field of quotients of the polynomial ring $F[x]$. Question. What are all the elements in $F(t)$ which are algebraic over $F$? I think ...
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4answers
33 views

Show that $\ker(\phi)$ is a maximal ideal if and only if $B$ is a field

Let $A$ and $B$ two commutative rings with unity $1_A \not= 0_A$ and $1_B \not= 0_B$. Consider $\phi : A \to B$ a ring epimorphism. Show that if $\ker(\phi)$ is a maximal ideal, $B$ is a field. I ...
2
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1answer
31 views

Can this extension of fields be transcendental?

Let $(R, \mathfrak m)$ be a local integral domain which is contained in a field $K$. Let $0 \neq x \in K$ be such that $\mathfrak m R[x]$ is a proper ideal of $R[x]$ (one can show for any $x$ that ...
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2answers
29 views

Is Z3 a sub-field of R?

The inverse numbers for the items in $\mathbb{Z3}$ are different than in $\mathbb{R}$ so I assume it's not a sub-field of $\mathbb{R}$. Am I correct? And in general, can sub-field of a infinite ...
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2answers
59 views

Product of two transcendental numbers is transcendental

let $\alpha,\beta$ be transcendental numbers.which of the following are true? 1)$\alpha\beta\ \text{ is transcendental}$. 2)$\mathbb{Q}(\alpha)\ \text{is isomorphic to }\mathbb{Q}(\beta)$ ...
3
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3answers
48 views

To find a field of $p^{2}$ elements ,where $p$ is prime

Show that there exists a finite field of $p^{2}$ elements for every prime $p\in\mathbb{N}$. What I thought is that if I find some irreducible polynomial of degree two over $\mathbb{Z_p[x]}$, then I ...
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0answers
14 views

Adjoining a separable element to a field makes the extension separable

Let $k$ be a field, and $K$ be an extension. Suppose $a\in K$ is separable over $k$. What's a clever way to show that $k(a)$ is separable (i.e., that all elements of $k(a)$ have separable minimal ...
4
votes
1answer
55 views

Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
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0answers
57 views

Isomorphism from $\mathbb{Q}(\sqrt{2})$ to $\mathbb{Q}[x]/\langle x^2 - 2\rangle$ [on hold]

I am just now beginning my first course in Fields. Sometimes I learn best by just being absolutely certain of some basic facts. This is why I like to ask simple True/False questions that I think are ...
1
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1answer
17 views

Finite Galois Field extension of a field $F$ containing all roots of unity

Let $F$ be a field that contains all roots of unity. Furthermore, let $K$ be a finite algebraic extension of $F$ with abelian Galois group . Then $$K= F(z_1,\ldots , z_n)$$ for some $z_i \in K$ ...
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2answers
19 views

Formally real field with two different orders

If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique. a) How different can be two (compatible with ...
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0answers
22 views

extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is $v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...
2
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2answers
24 views

Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. Find an element $Y \in K(X)$ so that $K(Y)=\{ f\in K(X) : \phi (f)=f\}$.

Let $K(X)$ be the field of rational functions of $X$ over some field $K$. Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. We have $L:=\{ f\in K(X) : \phi (f)=f\}$. Find ...
1
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1answer
16 views

An field extension of degree 2 is Normal Extension.

let $L\ \text{be a field and $K$ is extension of $L$ such that $[K:L]=2$ prove that $K$is normal extension} $ what i have tried is let $ f(x)$ $\text{be any irreducible polynomail in} $ $L[x] $ ...
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2answers
20 views

Degree of field extension using minimum polynomial

Let $K \subset L$ an algebraic extension. I want to prove for $a,b \in L$ that $$[K(b): K] \geq [K(a,b): K(a)],$$ Where for example $[K(b):K]$ is the degree of the field extension $K \subset K(b)$. ...
4
votes
2answers
78 views

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$? I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= ...
1
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1answer
31 views

What does it mean that “$f = g$ in $k(h(t))$?”

Let $k$ be a field and consider the rational function field $k(t)$. I was just reading that if $f(t),g(t),h(t)\in k(t)$ are such that $f(h(t)) = g(h(t))$, then "$f = g$ in $k(h(t))$." What does that ...
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votes
1answer
22 views

Constructible numbers defined over the rationals

If $z$ is constructible, then its minimal irreducible polynomial has a degree a power of $2$. Does the polynomial have to be defined over the rationals? I am asking this because we can ...
1
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1answer
66 views

Number of solutions in a field of order $32$ [duplicate]

Let $F$ be a field of order $32$. Then find the number of non-zero solutions $(a,b)\in F\times F$ of the equation $x^2+xy+y^2=0$. As , $|F|=32$ , so $(F\setminus\{0\},.)$ forms a group of order $31$, ...
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votes
1answer
41 views

field of fractions of $k[X]$ [on hold]

Let $k$ be a field and suppose $$k(X)=\text{field of fractions of }\ k[X]=\left\{ \frac{f(X)}{g(X)}\mid f,g\in k[X], g\neq 0\right\}.$$ Show that $k(X)$ is not a finitely generated $k$-algebra.
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2answers
24 views

Existence of elements of even order in a field with characteristic 2

I've read this statement in a presentation slide, but it isn't obvious to me on why this is true: Forgetting about BCH codes, the question is: if an alement $\beta$ has even order ($2k$ is always ...
1
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1answer
34 views

Basic question in Galois theory (on applying elements of the Galois group to a root of polynomial)

Suppose I have $K = \mathbb{Q}(\theta)$ and let $f$ be the minimal polynomial of $f$ over $\mathbb{Q}$. Suppose $f$ has degree $n$ so that the degree of $K$ over $\mathbb{Q}$ is $n$. Suppose further ...
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0answers
10 views

find dimension of Q[$x$]/<$(x^2+1)^2$> over Q?

I know that above is not a field because polynomial $(x^2+1)^2$ is reducible over Q. Then how to find out its dimension? i know it is $2$ if polynomial is $x^2+1$ as in this case field is isomorphic ...
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0answers
13 views

$k$-closed, defined over $k$, and pure inseparability

Let $\Omega$ be a large algebraically closed field, $k$ a subfield of $\Omega$, $\overline{k}$ the algebraic closure of $k$ in $\Omega$, and $k^i$ the field of purely inseparable elements over $k$. ...
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2answers
44 views

Number of solutions for the given equation in finite field of order 32.

Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$. I have figured out that non zero elements of this field forms a cyclic multiplicative group of ...
1
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1answer
39 views

Is $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$?

I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in ...
2
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1answer
39 views

Transposition not diagonalizable in characteristic 2

In another thread it was proved that transposition as a linear map is diagonalizable. This, however, does not hold when we are working over a field of characteristic 2. I suppose the proof of this can ...
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1answer
27 views

A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots

I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
0
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0answers
15 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: ...
11
votes
3answers
728 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
0
votes
1answer
81 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
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0answers
31 views

Regarding the number of isomorphisms between splitting fields

Let $\phi : F \rightarrow F_1$ be an isomophism of fields and $f(x) > \in F[x]$. Let $\Phi : F[x] \rightarrow F_1[x]$ be the unique ring isomorphism which extends $\phi$ and maps $x$ to ...
2
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0answers
21 views

How algebraically restrictive is the structure of $\Bbb Q_p$?

My question is a little difficult to explain, but I will try to state it. First, let me talk about the field $\Bbb R$ of real numbers. Let $K$ be the maximal real algebraic extension of $\Bbb Q$, that ...
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0answers
37 views

What does $F^n$ mean in the case of fields?

I am reading notes for Hilbert's Nullstellensatz and came across this expression. $$\exists F, \mathcal{V}_F(I):\{\xi \in F^n:f(\xi)=0, \forall f\in I\}\neq \emptyset$$ Here, $F=\mathcal{P}/m$ is a ...
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2answers
38 views

'Multiplying' by 0 in a field, field axiom proofs

The question says: The solution set was posted and there are a few things I don't quite understand from it. For the first one, I'm not entirely sure what's happening. It appears to be using the ...
2
votes
1answer
25 views

A question concerning archimedean places on number fields

Let $L$ be a number field and $G=Aut(L\mid \mathbb{Q})$. Let $|\cdot|$ be the usual archimedean value on $\mathbb{C}$ and, by abuse of notation, its restriction to $\mathbb{Q}$. Then the archimedean ...
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0answers
12 views

Linearly disjoint abelian extension of $\mathbb{Q}_p$

Given an abelian(finite or infinite) extension $K/\mathbb{Q}_p$ which is not the maximal abelian extension, can we always choose a cyclic extension $E=\mathbb{Q}_p(\zeta_n)$ such that $E\cap ...
0
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3answers
20 views

Show that $\{0\}$ and $V$ are the only linear subspaces of $V = k.$ where $k$ is a field.

On the surface this seemed easy, but my first attempt was rendered useless since I don't actually know that its an ordered field. I said: Let $W (\neq V)$ be a subspace of $ V$ then, if we let $v = ...
2
votes
1answer
31 views

Valuation ring between $F$ and $\mathcal O_F$

Let $(F,v)$ be a complete discrete valuation field (normalized) with ring of integers $\mathcal O_F$. Why cannot exist a valuation ring $A$ of $F$ such that $F\supsetneq A\supsetneq\mathcal O_F$ ...
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3answers
45 views

Suggestions for readings; Elliptic curves over function fields

I would love to know some good refercences about Elliptic curves over function fields. Especially in view with Mordell-Weil's Theorem. I am already familiar with the main proof of Mordell's theorem in ...
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1answer
25 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
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3answers
27 views

Show that for $p \neq 2$ not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square.

Show that for $p\neq2$, not every element in $\mathbb{Z}/p\mathbb{Z}$ is a square of an element in $\mathbb{Z}/p\mathbb{Z}$. (Hint: $1^2=(p-1)^2=1$. Deduce the desired conclusion by counting). So far ...
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1answer
24 views

Please find errors in my reasoning about field axioms

We can define a field F with the following properties: Binary operations + (addition) and ⋅ (multiplication) Commutativity Associativity Identities Inverses Distributivity Now, the additive ...
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2answers
26 views

Show that $K[X]/(P)$ is the splitting field of $P$.

Let $K$ a field and $P\in K[X]$ and irreducible polynomial. The fact that $K[X]/(P)$ is a field is fine. I want to show that it's the smallest field where that split $P$. First, let show that ...
0
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2answers
68 views

Why does the associative property of vector addition imply a sum may be written as $\alpha_1+\alpha_2+\cdots+\alpha_n$?

In an effort to understand that a sum involving a number of vectors is independent of the way in which these vectors are associated, I've tried to derive other bindings of certain vector additions in ...