# Tagged Questions

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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### 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 ...
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### Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
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### Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
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### (Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \$? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
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### Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
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### Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
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### The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$

I'm trying to solve this question: Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a field $F$. Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the ...
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### When the extensions Q(α,β) and Q(αβ) over Q are the same?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha\beta)$ over $\mathbb{Q}$ are the same. Thanks in advance.
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### Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals ...
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### Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the ...
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### I don't understand what a Pythagorean closure of $\mathbb{Q}$ is; how are these definitions equivalent?

I have two definitions of the said field. And frankly I don't see why one is equivalent to the other. It just doesn't add up. Let's look at wikipedia's definition. In algebra, a Pythagorean field ...
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### Proving that primes of the form $4k + 1$ are not Gaussian primes

Let $p$ be a positive integer prime such that $p\equiv1\pmod4$. Assume that $p$ is a Gaussian prime, so $(p)$ is a maximal ideal of $Z[i]$, and consider the field $K = Z[i]/(p)$. We state without ...
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### Number of homomorphisms from a stem field to a given field

This is a homework, but I've generalized it as possible in order not to have exact answer rather that to understand the very principle of solution. The problem is following, for a given irreducible ...
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### Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub ...
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### Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
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### 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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### Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $< \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
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### Galois extension of $\mathbb C$ is also Galois over $\mathbb R$??

Is any Galois extension of $\mathbb C$ also Galois over $\mathbb R$? I know if that extension is finite, Then it is true because $\mathbb C$ is algebraically closed. But How about the infinite case? ...