All Questions
6
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Confusing example of prime and irreducible elements from my lecture script in abstract algebra [duplicate]
Could you please help me to understand the following "example" from my lecture script in the abstract algebra?
Example 12.34. Let $R = K[[x]]$ be a formal power series ring over a field $K$....
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3
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3
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Is $\mathbb{Z}[√13] $ a Unique factorization domain?
I think it is not so as $12 $ can be written in two ways $12=2.6=(1+\sqrt{13})(-1+\sqrt{13})$. Are these two factorization unique upto irreducibles? Please help.
- 241
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$[K(\beta):K] \ \deg{f_i} = [K(\alpha): K] \ \deg{g_i}$
Let $f$,$g$ be irreducible polynomials over the field $K$, both without multiple zeros (in an algebraic closure of $K$). Let $L=K(\alpha)$ and $E=K(\beta)$ be extensions of $K$ with $f(\alpha)=0$ and $...
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35
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What are examples of irreducible but not prime elements?
I am looking for a ring element which is irreducible but not prime.
So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$.
This is irreducible because in any ...
- 1,057
7
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1
answer
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Irreducibility of an integer polynomial with exponents in linear sequence?
Let $b$ and $n$ be two positive integers. Is there are a general result which tell us when the polynomial $$1+x^{b}+x^{2b}+x^{3b}+\cdots+x^{nb}$$
is irreducible over the integers?
I know that $$1+x+...
user123641
2
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3
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Number of solutions of polynomials in a field
Consider the polynomial $x^2+x=0$ over $\mathbb Z/n\mathbb Z$
a)Find an n such that the equation has at least 4 solutions
b)Find an n such that the equation has at least 8 solutions
My idea is to ...
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