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
votes
3answers
329 views

Show that $f(x) = x^p -x -1 \in \Bbb{F}_p[x]$ is irreducible over $\Bbb{F}_p$ for every $p$.

a) Show that $f$ has no roots in $\Bbb{F}_p$ Let $F^*$ be the multiplicative group of $\Bbb{F}_p$. Then, by lagrange's thoerem for all nonzero $a \in \Bbb{F}_p$, $x^{p-1} = 1 \implies x^p=x ...
0
votes
1answer
194 views

Isn't $(0)$ a prime ideal in a field?

I have read in multiple places that a field $K$ has a Krull dimension of $0$. How is this true? Isn't $(0)\subset K$ a prime ideal in $K$? Obviously $K$ is an integral domain. Thanks in advance!
1
vote
1answer
119 views

Problem about intermediate fields in the extension

Let $E,K$ be intermediate fields in the extension $L/F$ (a) If $[EK:F]$ is finite, then $$[EK:F] \leq [E:F][K:F] $$ (b) If $E$ and $K$ are algebraic over $F$, then so is $EK$ For (a), I try two ...
6
votes
2answers
257 views

Extension of residue fields and algebraic independence

Let $A$ be a Noetherian integral domain, $B$ a ring extension of $A$ that is an integral domain, $P \in \operatorname{Spec} B, \, p = P \cap A$. Denote by $\kappa(p),\ \kappa(P)$ the residue fields of ...
48
votes
12answers
4k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
3
votes
3answers
111 views

isomorphic polynomial rings

I'm certain that this is a dumb question, but I'll ask anyway. I know that if $\theta : F \to K$ is a field isomorphism then we get an induced isomorphism $\varphi:F[x] \to K[x]$ such that $\varphi|F ...
2
votes
1answer
31 views

Are composite fields unique?

Suppose for $i=1,2$ that $\Omega_i$ is a field containing fields $K_i$ and $L_i$, with $K_1 \cong K_2$ and $L_1 \cong L_2$. Is it then true that there is an isomorphism $K_1L_1 \cong K_2L_2$ of ...
3
votes
2answers
223 views

polynomial rings — inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
2
votes
1answer
57 views

Extensions of number fields

Let $L|K$ be an extension of number fields and consider the corresponding (integral) extension of ring of integers: $R_L|R_K$. Note that $R_L$ and $R_K$ are finitely generated over $\mathbb{Z}$, hence ...
1
vote
0answers
36 views

$A$ is similar to $B$ in $E$ if only if $A$ is similar to $B$ in $F$. [duplicate]

If $E$ is a field, $F/E$ is a field extension, let $A$ and $B$ be two matrices with entries in $E$ then $A$ is similar to $B$ in $E$ if only if $A$ is similar to $B$ in $F$. I think it's true, but I ...
1
vote
1answer
94 views

Intersection of algebraic field extensions.

Let $K \subset L$ be an algebraic field extension, and let $\alpha$ and $\beta$ be non-conjugates elements of $L$. Is it true that $K(\alpha) \cap K(\beta) = K$? This is probably easy but I just can't ...
7
votes
1answer
125 views

Extension degree of residue field.

Let $k$ be a field, and $A$ be a finitely generated $k$-algebra with $\text{dim}(A)\leq 1$. Then for any maximal ideal $\mathfrak{m}$ of $A$, does this inequality $[A/\mathfrak{m}:k]<\infty$ hold? ...
6
votes
4answers
678 views

Does there exist a field which has infinitely many subfields?

Does there exist a field which has infinitely many subfields? Does there exist an enormous supply of such fields? I don't know how to begin.
2
votes
2answers
164 views

Finite field extension over $\mathbb F_2$

I don't see why $[L:K]=4$, where $L = \mathbb{F}_2(x,y) = \operatorname{Quot}(\mathbb{F}_2[x,y])$ and $K = \mathbb{F}_2(x^2,y^2) = \operatorname{Quot}(\mathbb{F}_2[x^2,y^2])$ Let $p(X) = X^2-x^2 ...
2
votes
1answer
88 views

Splitting field of resolvent equals that of $f$

Lemma: Let $\Psi \in k[X_1,...,X_n]=:B$ be s.t. $stab_{S_n}(\Psi)=H \subset S_n$, $S_n/H=\{ \Psi, t_2 \Psi,..., t_e\Psi \}$, $\Delta_\Psi$ the discriminant of $L_\Psi:=\prod_{i=1}^e (X-t_i \Psi)$, and ...
1
vote
1answer
152 views

Splitting field of $x^m - 1$ over $\mathbb F_p$

I need to find the splitting field of a polynomial $x^m-1 \in\mathbb{F}_p[x]$. I know that if $(m,p)=1$ then the splitting field is $\mathbb{F}_p(z)$ where $z$ is primitive root of unity of order $m$. ...
3
votes
4answers
142 views

Finitely generated field extensions

This is a really dumb question, but why is $\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2} : a,b \in \Bbb{Q}\}$? I am having trouble writing field extensions in this way.
0
votes
1answer
46 views

A morphism which fixes one root of an irreducible polynomial must also fix the others.

Let $E/K$ be a field extension, let $p(x)$ be an irreducible polynomial in $K[x]$ which splits in $E$ with roots $\alpha_1$, $\alpha_2$, etc., and let $\sigma$ be an automorphism of $E$ which fixes ...
4
votes
2answers
79 views

If $1+g(x)^2$ has an irreducible factor of odd degree in $F[x]$, then there is some $a\in F$ such that $a^2 =-1$

Let $F$ be a field and $g(x)$ in $F[x]$. Prove if $1+g(x)^2$ has an irreducible factor of odd degree then there exists $a$ in $F$ such that $a^2 =-1$. I didn't get too far on this problem. It doesn't ...
2
votes
1answer
174 views

Minimal polynomial over an extension field divides the minimal polynomial over the base field

I need help proving this theorem: Given the field extension: $\mathbf{K} \subseteq \mathbf{L}$, for $\alpha \in \mathbf{L}$ and $g(x) \in \mathbf{K}[x]$, $\alpha$'s minimal polynomial over $K$, ...
3
votes
2answers
566 views

source to learn Galois Theory

What kind of recommendations do you have for a very good source to learn Galois Theory? Is there any Atiyah-MacDonald-type book on Galois theory? What is your opinion on the chapters from Lang and ...
0
votes
1answer
37 views

Factorization of a polynomial in its splitting field.

Let $K|k$ be a extension field (i.e, $K|k$ denotes that $k \subseteq K$, where both $K,k$ are fields) with $K$ the descomposition field of $p(x)\in k[x]$. I don't get the following: If in the ...
14
votes
2answers
301 views

Sum of irrational numbers, a basic algebra problem

Let $x_1,\dots,x_n$ be positive rational numbers. If $\sqrt[l_1]{x_1},\dots,\sqrt[l_n]{x_n}$ are all irrational numbers (where $l_1,l_2,\dotsc,l_n\in\Bbb N^*$), does it follow that $$\sqrt[l_1]{x_1}+ ...
1
vote
1answer
55 views

In this theorem from Lang's Algebra, what is the codomain of this map?

I need help understanding what the final sentence below means. Paraphrased from Lang's Algebra: Let $K/k$ be Galois and $F$ an arbitrary extension, $K,F$ subfields of some other field. Then $KF/F$ ...
0
votes
2answers
52 views

Field homomorphism respects arbitrary arithmetic expression

A field homomorphism $f:A \to B$ respects the fields' binary operations, in the sense that $f(x + y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$. When you have an explicit expression like $expr = x^3 + 15x ...
0
votes
1answer
23 views

Elements $a,b\in L$ of degree $3$ over $K$ such that $a+b$ has degree $6$ over $K$

A finite field extension $L/K$ and elements $a,b\in L$ of degree $3$ over $K$ such that $a+b$ has degree $6$ over $K$. Can you give me an example about such field extension? Thanks.
0
votes
2answers
226 views

Square roots in arbitrary fields

I'm a little confused about a certain argument concerning square roots. The problem is Dummit and Foote, 13.2.9., detailed here with a solution also given. Specifically my problem is as follows: ...
2
votes
1answer
102 views

Fields $k$ with subrings whose quotient field is $k$

Let $k$ be a given field, any field. Is there a subring $A$ of $k$ such that $k$ is the quotient field of $A$? Let's restrict ourselves to fields I know anything about; subfields of $\mathbb{C}$, ...
4
votes
1answer
162 views

Infinite-dimensional extensions of $\mathbb Q$

I need help to solve the following exercise: Let $X$ be an indeterminate over $\mathbb Q$ (so a transcendental number) and consider the field extensions $\mathbb Q\subseteq \mathbb ...
0
votes
1answer
50 views

What is $\mathbb Q (\theta)$ where $\theta$ is transcendental?

Let $\theta \in \mathbb R$ be irrational. Is $\mathbb Q(\theta) = \mathbb R$? What does $\mathbb Q (\theta)$ "look like"?
3
votes
4answers
241 views

Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$

Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$? That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?
0
votes
1answer
41 views

Field extension $k(a)$

Well, first I write some definitions: Let $K|k$ be a field extension. Then $k(a)$ denotes $$ k(a)=\bigcap\{F:k\subseteq F \subseteq K \,\ a\in F \} $$ and is the smallest field of the ...
-1
votes
1answer
55 views

A subfield of $\mathbb C$ with a real imbedding and finite extension

If $\mathbb C/K$ is a finite extension of fields and there is an imbedding from $k$ to $\mathbb R$, my conjecture is $[\mathbb C:k]=1$ or $2$. I need some idea. Thank you.
2
votes
2answers
79 views

About the notation $\mathbb{F}(\theta)$ in the field extension.

Actually, I've not studied deeply field theory and I'm reading an article about the field theory, especially field extension. Suppose that $\mathbb{F}$ be a field. And the book treats ...
9
votes
1answer
348 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
1
vote
0answers
51 views

Algebraic closure of rational fractions

I work with matrices whose entries are rational fractions of polynomials $$ M(x) = \left( \begin{array}{ccc} R_{11}(x) & \dots & R_{1n}(x) \\ \vdots & \ddots & \vdots \\ R_{n1}(x) ...
4
votes
1answer
103 views

Possibilities for $[KL:F]$ when $[K:F]=[L:F]$ is prime

Suppose $K/F$ and $L/F$ are extensions of $F$ (contained in some common field) of degree $p$, where $p$ is prime. Standard arguments show that $[KL:F]$ must be in $\{p,2p,\ldots,p^2\}$. But are all ...
1
vote
1answer
58 views

Factorisation of $x^{4}-x^{2}+2x+1$ in $\mathbb Q[x]$?

Hallo can any one tell me what is the idea behind this? the polynomial $p=x^{4}-x^{2}+2x+1 \in \mathbb Q[x]$ is irreducible, because in in $F_{2}[x]$ it can factored to $(x^2+x+1)^2$ and in ...
3
votes
2answers
93 views

About the number of $K$-homomorphisms

Let $K\subseteq L\subseteq M$ be fields with $[L:K]=n$. I should prove that the number of $K$-homomorphisms from $L$ to $M$ (so field homomorphisms that fix pointwise $K$) is less or equal than $n$. ...
-1
votes
1answer
52 views

Algebraic closure of the field of rational fractions

I was wondering if the field of rational fractions of the variables $x_1,...,x_n$ is algebraically closed? Thanks for your help.
5
votes
0answers
197 views

Extension fields of $\mathbb Q$

Let $\mathbb Q$ be the field of rationals. Let $m_1, m_2,\dots, m_k$ be in $\mathbb N^*$. Let $t_1, t_2, \dots, t_k$ be in $\mathbb N$. Suppose $t_i^{1/m_i}$ $\neq $ $q t_j^{1/m_j}$ for ...
5
votes
0answers
162 views

Purely inseparable extension of algebraic function field

Let $K$ be a field with $\operatorname{char}(K) = p > 0$ and with the property that $[K:K^p] < \infty$. Let $F/K$ be an algebraic function field, i.e there is an element $x \in F$ which is ...
4
votes
1answer
242 views

$K(u,v)$ is a simple extension of fields if $u$ is separable

I have problems to prove the following statement. Let $K$ be a field and let $K(u,v)$ be an algebraic extension of $K$. If $u$ is separable over $K$ then $K(u,v)$ is a simple extension. (My ...
0
votes
2answers
193 views

If F is a finite field, then $F^*$ is cyclic and $F=\Bbb{Z}_p(\alpha)$ for some $\alpha$.

From Galois Theory (Rotman): If F is a finite field, then $F^*$ [which is the multiplicative group] is cyclic and $F=\Bbb{Z}_p(\alpha)$ for some $\alpha$. Proof If $|F|=q$, take $\alpha$ to ...
2
votes
1answer
57 views

What does $K^{1/p}$ for a field $K$ mean?

In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows. $k$ is a ...
0
votes
1answer
118 views

Problem: when the sum of two squares is a square

Please, I need help to solve the following problem: Let $K$ be a field with characteristic different from $2$ and $3$. Show that the following statement are equivalent: The sum of two ...
8
votes
1answer
142 views

Question about the Galois extension of a given field extension

Let $K=\mathbb{Q}(\omega)$ be given, where $\omega^3=1$. I want to know: (1) Whether there is a Galois extension $L/\mathbb{Q}$ containing $K$ such that $\mathrm{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_4$? ...
26
votes
6answers
1k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
4
votes
2answers
115 views

If $f\in\mathbb{F}_p[x]$ is irreducible and has a root in $\mathbb{F}_{p^n}$, then $f$ splits over $\mathbb{F}_{p^n}$ [duplicate]

Let $f(X) \in \mathbb F_p[X]$ irreducible with $p$ prime and assume $\exists \alpha \in \mathbb F_{p^n}: f(\alpha) = 0$ where $n \geq 1$. I then have to prove that $f$ splits over $\mathbb ...
11
votes
1answer
172 views

Are the real numbers a nontrivial simple extension of another field?

Is there a proper subfield $K$ of the real numbers and a real number $\theta$ such that $\mathbb R = K(\theta)$? I thought of this question earlier idly wondering about what the structure of the ...