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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4answers
29 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
30 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
28 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 ...
1
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2answers
56 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
votes
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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votes
0answers
13 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
48 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
56 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
16 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 $z_i \in K$ ...
1
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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 ...
1
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0answers
18 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
votes
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
vote
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] $ ...
1
vote
2answers
19 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
vote
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 ...
0
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
65 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.
1
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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
vote
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 ...
0
votes
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 ...
0
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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 $k$. ...
0
votes
2answers
43 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
vote
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
votes
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 $\overline{\overline{L}}=\overline{L}$. ...
1
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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
votes
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 ...
0
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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 ...
1
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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
votes
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 $F\supsetneq A\supsetneq\mathcal O_F$ ...
2
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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 ...
0
votes
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 ...
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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 ...
1
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1answer
21 views

Does the splitting field of an irreducible polynomial contain all extensions over which the polynomial factors?

Say $f$ is an irreducible polynomial with coefficients in a field $F$. Say $f$ is no longer irreducible over some extension $K$ of $f$, i.e. $f$ factors into a product of (irreducible) polynomials ...
2
votes
2answers
47 views

If a field element is simultaneously an $m$th and $n$th power… [closed]

Suppose $F$ is a field. Suppose we have $a,b \in F $ and relatively prime integers $m,n \geq 1$ such that $a^m = b^n$. Can I conclude that there is some $c \in F $ such that $c^{mn} = a^m = b^n$?
4
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0answers
38 views

Algebraic extension of perfect field is algebraically closed

Let $F$ be a perfect field, i.e. every irreducible polynomial over $F$ has distinct roots in the algebraic closure of $F$. Suppose that $K$ is an algebraic extension of $F$ with the property that ...
0
votes
1answer
41 views

Proof of equality of $2$ polynomials without using Galois Theory?

I have the following situation and ask myself whether an argument is available to prove the equality of the $2$ following polynomials. Both polynomials are of degree $n$ and are monic. The ...
4
votes
1answer
56 views

Algebraic closure with no nontrival automorphism

In Milne's notes on Galois theory, Chapter 7, p.91 he remarked that it is consistent without the axiom of choice that there exists an algebraic closure $L$ of $\mathbb{Q}$ with no nontrivial ...
1
vote
1answer
28 views

If $g(x) \in K[x]$, the $g(\alpha)=0$ if and only if $f(x)|g(x)$

Let $K $ a subfield of $\mathbb{C}$, $\alpha$ a complexe number which is algebraic on $K$ and $f(x) \in K[x]$ the minimal polynomial of $\alpha$ on $K$. If $g(x) \in K[x]$, the $g(\alpha)=0$ if ...