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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
126 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
184 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 $\mathrm{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_4$? ...
27
votes
6answers
2k 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
151 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
195 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 ...
2
votes
1answer
102 views

Given fields $F\subset K \subset E$ with $E/F$ normal, prove that $E/K$ is normal

If $F\subset K \subset E$, with $E/F$ normal, prove that $E/K$ is normal. I know that $F\subset K \subset E$ can imply if $E/F$ is algebraic then $E/K$ is algebraic. But how the 2-nd condition ...
3
votes
2answers
173 views

Subgroup with Finite Index of Multiplicative Group of Field

Let $F$ be an infinite field such that $F^*$ is a torsion group. We know that $F^*$ is an Abelian group. So every subgroup of $F^*$ is a normal subgroup. My question: Does $F^*$ have a proper ...
15
votes
1answer
395 views

A *finite* first order theory whose finite models are exactly the $\Bbb F_p$?

Since this question turned out to be trivial, I'm now asking this strengthened version: Is there a finitely axiomatized first order theory $T$ in the language of rings such that its finite models ...
7
votes
1answer
155 views

A first order theory whose finite models are exactly the $\Bbb F_p$

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of ...
2
votes
2answers
72 views

Simple question about splitting fields

Let f(x) be a polynomial in F[x]. Let K be the splitting field of f over F. Let a be an element of F. Is K also the splitting field of (x-a)f(x)? I think it should be. I just want to make sure I'm ...
4
votes
2answers
124 views

Characterizing continuous exponential functions for a topological field

Given a topological field $K$ that admits a non-trivial continuous exponential function $E$, must every non-trivial continuous exponential function $E'$ on $K$ be of the form $E'(x)=E(r\sigma (x))$ ...
7
votes
1answer
158 views

Lower Bound on Number of Wildly Ramified Extensions of $\mathbb{Q}_p$

Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. ...
3
votes
1answer
72 views

When a subfield of a splitting field is a splitting field

Look at the following proposition: Let $K\subset L\subset M$ be three fields. If $M$ is a splitting field over $K$ of a polinomial in $K[X]$ and moreover if for every $\sigma\in G=Gal(M/K)$ we ...
5
votes
1answer
168 views

Valuation ring in an algebraically closed field

Let $k$ be an algebraically closed field and $(R, \mathfrak{m})$ a valuation ring in $k$ i.e. the field of fractions of $R$ is $k$. Then Mumford (Red Book, page 127) claimed that the residue field ...
0
votes
1answer
33 views

Prime field of a skew field is a field

I'm having a question about skew fields (or division rings). The intersection of all the non-trivial sub skew fields is called the prime field. Now it says that this prime field is a field because it ...
1
vote
1answer
54 views

Finding $\operatorname{gal}(f\mid\Bbb C(\omega))$ and $\operatorname{gal}(f\mid\Bbb R(\omega))$

How to solve following: Let $\Bbb C(\omega)$ be a field of rational functions with one undetermined $\omega$ over the field of complex numbers $\Bbb C$. If $f(x)=x^5+\omega$, find a) ...
4
votes
2answers
189 views

Dummit and Foote p.544, Proposition 31 regarding constructing an algebraic closure.

I have a question on the proof of the following proposition in Dummit and Foote (3rd edition, Proposition 31, p.544). This section is what I entirely quote from the book. Proposition. Let $K$ be ...
0
votes
0answers
49 views

How many irreducible factors of grade $6$ there is in $\mathbb{F}_{2}\left[ x\right]$? [duplicate]

How many irreducible factors of grade $6$ there is in the polynomial ring $\mathbb{F}_{2}\left[ x\right]$? I have solved this by using the fact that every irreducible polynomial of grad $i$ is a ...
5
votes
0answers
71 views

Matrix representation of field automorphism

Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= ...
5
votes
1answer
68 views

If a monic $f\in\overline{K}[x]$ has a power $f^n\in K[x]$, where the characteristic of $K$ doesn't divide $n$, then must $f\in K[x]$?

Suppose you have a monic polynomial $f(x) \in \overline{K}[x]$, and some integer $n>1$, where $\mathrm{char}(K)\nmid n$, and $\big (f(x)\big )^n\in K[x]$. Does it imply $f(x) \in K[x]$? The ...
5
votes
1answer
538 views

A subset of a field that is a subfield

It can be verified that the following assertion is true: a subset $S$ of a field $F$ is a subfield if $S$ contains the additive and multiplicative identities 0 and 1, if $S$ is closed under addition, ...
1
vote
1answer
70 views

How many solutions does this equation has in a finite field

I'm working in a finite field $F_q$ where q is a primepower. As a small part of a problem I'm working on, I have to find how many solutions the equation $x^2-ay^2-1=0$ has in $F_q$, with $a \in ...
0
votes
2answers
155 views

Calculate $irr(a,\mathbb{Q})$ where $a=\sqrt{2}+\sqrt{3}\in\mathbb{R}$ [duplicate]

Consider the element $a=\sqrt{2}+\sqrt{3}\in\mathbb{R}$. Calculate $irr(a,\mathbb{Q})$. What I did: Calculate powers of $a$. $a^2=5+2\sqrt{6},a^3=11\sqrt{2}+9\sqrt{3},a^4=49+20\sqrt{6}$. I wish to ...
3
votes
2answers
99 views

Isomorphic multiplicative groups of quadratic extensions

What are all ordered pairs $(n,m)$ such that the multiplicative groups of the fields $\mathbb{Q}(\sqrt{n})$ and $\mathbb{Q}(\sqrt{m})$ are isomorphic? I saw a question earlier today claiming that ...
4
votes
1answer
56 views

Field extensions of $\prod \Bbb F_p /U$

The ultraproduct of all finite prime fields $ \Bbb F_p $ (over a nonprincipal ultrafilter U) is a field of characteristic 0. How do I show that it has exactly one extension of degree n for each ...
8
votes
4answers
258 views

Why is $\{a + b\sqrt2 + c\sqrt3 : a\in\Bbb{Z}, b, c \in\Bbb{Q}\}$ not closed under multiplication?

The set $R = \{a + b\sqrt{2} + c\sqrt{3}: a \in \Bbb{Z}, c, b \in \Bbb{Q}\}$ is not closed on multiplication, my textbook states. Why is this? And related to that: why then is $S = \{a + b\sqrt{2} : ...
4
votes
1answer
366 views

which of the following is/are algebraic over rationals

which of the following is/are true? $\sin 7^\circ$ is an algebraic over $\mathbb{Q}$ $\sin^{-1}(1)$ is algebraic over $\mathbb{Q}$ $\cos (\pi/7)$ is algebraic over $\mathbb{Q}$ $\sqrt{2}+\sqrt{\pi} ...
2
votes
1answer
59 views

$a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$.

I want to find a $a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$. Any ideas? Thanks.
3
votes
2answers
116 views

Separability of polynomials

From my textbook (Robert Ash's Basic Abstract Algebra, section 3.4): 3.4.2 Proposition If $$f(X)=a_0 + a_1X + \dots + a_nX^n \in F[X],$$ let $f'$ be the derivative of $f$, defined by ...
7
votes
2answers
135 views

Showing $\mathbb{Q}(2^{1/3}+2^{1/5})=\mathbb{Q}(2^{1/3},2^{1/5})$

In order to prove $[\mathbb{Q}(2^{1/3}+2^{1/5}):\mathbb{Q}]=15$, I want to show $\mathbb{Q}(2^{1/3}+2^{1/5})=\mathbb{Q}(2^{1/3},2^{1/5})$. Any suggestions?
3
votes
1answer
322 views

basic problem about field extension and irreducibility of polynomial

If $\alpha_1, \cdots, \alpha_n$ are distintct roots of a polynomial $p(x)$. If I want to show that $p(x)$ is irreducible over a field $F$, is it suffices to show that $deg (p) \leq [F(\alpha_1, ...
2
votes
2answers
74 views

Proof that field extensions of degree 3 over $K$ with $\mathrm{char}(K) \neq 3$ are separable

My question is how one can prove, for all field extensions $K \subset L$ with $[L:K]=3, $ char($K$) $\not=3$, that $L$ is separable over $K$. I understand this proof with 2 in stead of 3. I ...
8
votes
2answers
1k views

Show that an algebraically closed field must be infinite.

Show that an algebraically closed field must be infinite. Answer If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F ...
0
votes
1answer
532 views

Radical extensions

Suppose we have $$ K \subset K(a_1) \subset K(a_1, a_2) \subset K(a_1, a_2, a_3) $$ such that $a_j^{p_j}\in K_{j-1}$: a radical extension in other words. I am having trouble understanding why for ...
2
votes
1answer
78 views

Prove that this splitting field has degree 4 over $\Bbb{Q}$.

If $m$ and $n$ are distinct square-free positive integers greater than $1$, show that the splitting field $\Bbb{Q}(\sqrt{m}, \sqrt{n})$ of $(X^2-m)(X^2-n)$ has degree 4 over $\Bbb{Q}$. Proof ...
2
votes
1answer
165 views

On the fundamental theorem of field extensions

I'm re-reading the fundamental theorem of field extensions. (K is normal $\iff$ K is a factorization field.) Assume $K=F(\alpha_1, \dots , \alpha_n)$, is the factorization field of $f\in F[x]$, over ...
5
votes
3answers
2k views

Understanding examples of subfield and prime subfield of a finite field

I have already taken a look at this answer. Somehow it did not answer my question. As I can find, in various literatures, A lecture note, Definition 4.1: Let $F$ be a field. A subset $K$ that is ...
1
vote
1answer
108 views

Finite fields as splitting fields

hey guys so i stumbled upon an example that begins with "Consider GF(25). This can be constructed as the splitting field of $t^2 - 2...$" But the theorem states that it is the splitting field of the ...
0
votes
1answer
64 views

Primitive element (fields)

I'm re-reading the primitive element lemma and I can't reason the following concept. Let $f,g\in F[x]$ be in the polynomial ring of one variable over the field $F$. Let those two polynomials have a ...
3
votes
0answers
95 views

Is there a more efficient way of proving a polynomial is primitive?

I have found an impressive Java implementation of a $ℤ2$ Polynomial class in which there seems to be an efficient implementation of a test for reducibility (see ...
6
votes
3answers
2k views

Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$

Determine all the subfields of the splitting fields of this polynomial. I chose this problem because I think to complete it in great detail will be a great study tool for all of the last chapter, as ...
9
votes
2answers
252 views

This tower of fields is being ridiculous

Suppose $K\subseteq F\subseteq L$ as fields. Then it is a fact that $[L:K]=[L:F][F:K]$. No other hypotheses are needed (I'm looking at you, Hungerford V.1.2). Now obviously ...
4
votes
1answer
226 views

Degree of splitting field extensions

The problem states: Let $f (x) = x^3+px+q$ be an irreducible cubic polynomial with rational coefficients and let $K$ be the splitting field of $ f(x) $ over $\mathbb{Q}$. Prove that $ [K : ...
8
votes
3answers
521 views

Do maximal proper subfields of the real numbers exist?

To clarify the problem, consider the field ${\Bbb R}$ as a field extension of ${\Bbb Q}$ using some sort of Hamel basis. Does there exist a field $F\subsetneq{\Bbb R}$ such that $F(\sqrt{2})={\Bbb R}$ ...
7
votes
0answers
98 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
2
votes
1answer
93 views

Non-archimedean matrix fields

Are there any examples of sets $A\subseteq{\Bbb C}^{n\times n}$ for which you can find operations so that $(A,+,\,\cdot\ ,<)$ is a non-archimedean ordered field? I feel like the answer is probably ...
2
votes
1answer
502 views

Methods to show polynomials are irreducible

I would like to show that $x^3 + x^2 - 2x - 1$ is an irreducible polynomial over $\mathbb{Q}$. What are my standard lines of attack to solve this problem? Typically I go to Eistenstein, but it does ...
4
votes
1answer
66 views

Comparing fields with same degree

Two part question: Are the fields $\mathbb{Q} (\sqrt[3]{2}, i \sqrt{3})$ and $\mathbb{Q} (\sqrt[3]{2}, i, \sqrt{3})$ identical in algebraic structure? I have in notes that they both have degree of 6 ...
1
vote
2answers
260 views

Is any homomorphism between two isomorphic fields an isomorphism?

Is any homomorphism between two isomorphic fields an isomorphism? What I mean is that two fields are called isomorphic if there exist one homomorphism between them . But not ...
3
votes
2answers
111 views

If $\alpha= (1+\sqrt{-19})/2$ then any ring homomorphism $f : \mathbb{Z}[\alpha] \rightarrow \mathbb{Z}_3$ is the zero map

This is from a past qualifying exam. Here is the question: If $\alpha= (1+\sqrt{-19})/2$ then any ring homomorphism $f : \mathbb{Z}[\alpha] \rightarrow \mathbb{Z}_3$ is the zero map. Here is ...