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Abstract Nonsense 
I just recently stumbled upon your blog, and I’d just like to say that I enjoy it very much.
Keep up the good work,
Justin Gilmore
i just had a question
that if we have a reducible polynomial, from R=R[x] , polynomials over real numbers and we have a quotient ring R/f(x)R with this particular reducible f(x), how can we show that this quotient ring would not be a field since f(x) is reducible, can we come up with a contradiction somehow that an inverse of f(x) would not exist?
thanks
In the quotient ring, f(x) = 0, so it’s trivial that f has no inverse. But to find a nonzero number without an inverse, write f(x) as g(x)h(x); you can do that since f is reducible. Now g(x) and h(x) are nonzero, but g(x)h(x) = 0, so that they’re zero divisors. No zero divisor can have an inverse, because if there exists p(x) with g(x)p(x) = 1, then it means that h(x) = 1*h(x) = g(x)h(x)p(x) = 0, a contradiction.
thanks, and i had another question.
how can i show that the set of real numbers is a subfield of R/f(x)R
i know that the real numbers itself is a field, and that R/f(x)R is closed under addition and multiplication, so is it enough to show that if multiplicative inverse and 0 exist then it would be a subfield?
and from the previous proof, why are we saying that g(x)h(x)=0? it has to be f(x) which can be nonzero right?
Since we’re modding out by f(x), it means that f(x) = 0. Think of it this way: if we want to construct the complex numbers, i.e. C, then we’ll mod out by x^2 + 1. Then x = i, and indeed i^2 + 1 = 0.
You’re assuming f(x) is reducible, so we can express it as g(x)h(x).
Finally, the real numbers arise as a subfield of R[x]/f(x) by taking the subring of constants. The sum, product, additive inverses, and reciprocals of constants are all constants, regardless of what we’re modding out by.
but why are we modding it out by f(x),if we had have the field R/f(x) we could have done mod f(x) but our field is R/f(x)R, so we shud mod by f(x)R, but then again thats a set, so we cant do that, but i still dont get the reason to do mod f(x)?
Oh, I’m just abusing notation. Modding out by the ideal f(x)R[x] can be written as “modding out by f(x),” just like modding out by the ideal (2) in Z is often written as “modding out by 2.”
can we do this using a specific example, say f(x) = x^2 + 1
Well, not if you want a reducible f… but try f(x) = x^4 + 1, which for a while I was sure was irreducible, go figure. f breaks down as (x^2 + SQRT(2)x + 1)(x^2 – SQRT(2)x + 1). Then, if we let j be the image of x in the quotient ring, we get that j^2 + SQRT(2)j + 1 is not a unit, or else we eventually get 1 = 0.
If a quartic is too offputting, let f(x) = x^2 – 1…
hey
if R is a ring and r is an element of R then rR is always a subring?
how can we prove this?
rR is an ideal; it’s a subring if you allow rings not to have 1. To prove that, note that rR is closed under the ideal operations because ra + rb = r(a + b), ra – rb = r(a – b), bra = rab.
how can we show that for a field F, F-> F[x]/(x), can be a surjective mapping?
In F[x]/(x), every f(x) = a(n)x^n + … + a1x + a0 is equivalent to a0. So given any f(x) in F[x]/(x), a0 in F maps to it.
prove that if n divides p – 1 then the congruence x^n – 1 congruent to 0 mod p, has exactly n solutions
can be solve the congruences x^2 = 1 mod 7
x^2 = 1 mod 13, for x. using chinese remainder theorem.
and are these sol unique?.
and hence is it true for all prime numbers.
Nida, just use x = (+/-)1 mod 7 and mod 13.
Tanya, give me a day or two; this is relevant to my research, so once I get around to explaining it, I’ll prove that x^n = 1 mod p has n solutions iff n divides p-1.
Tanya:
Since Z_p is a field, x^n – 1 has at most n solutions.
x^n-1 = 0 iff x^n = 1, so by looking at the multiplicative group U(Z_p), we’re looking for elements of order dividing n. Since U(Z_p) is cyclic, we reach the needed result (if G is cyclic of order m, and k divides m, look at the set of all g^{m/k}.)
If f(x) is an element on F[x], polynomial ring, and f(x) has n disticnt roots,
>in its splitting field E, then galois(E/F) is isomorphic to a subgroup
>of the symmetric group Sn.
as far as i know is that :
The elemnts of Gal(E/F) are the F-automorphisms of E.Any
h in Gal(E/F) sends roots of f(x) into roots of f(x)
but where do go from here?
Could you help me with this very standard problem in Abs Alg:
Find the Glaois Group of x^8 – 1 over Q.
Okay… x^8 – 1 factors as (x^4 + 1)(x^2 + 1)(x + 1)(x – 1). The last two factors can be ignored. The second factor splits as (x + i)(x – i) in Q[i], where the first factor splits as (x^2 + i)(x^2 – i). Since -1 is a square in Q[i], it means that if we have a square root of i, we have a square root of -i. So the splitting field has degree 2 over Q[i], hence degree 4 over Q[i]. To see that the Galois group is V rather than Z/4Z, it’s enough to find 3 quadratic subfields (or even 2). Q[i] is obvious. Less obvious is Q[SQRT(2)], since SQRT(i) = (1 + i)/SQRT(2) and SQRT(-i) = (1 – i)/SQRT(2), so adding them we get SQRT(2). Likewise, subtracting them we get SQRT(-2), hence Q[SQRT(-2)].
The next time you though indirect- & unconsciously threaten me, God’s attorney’s just like that welcome+ to take your motivated opinions about ‘it’ up with me, so that I can more easily find out, what we motivatedly owe each other & e.g. ourselves, is there anything, I don’t know, if I can explain to myself, it’s a false attitude & This Buddhism, I ‘hopefully’ seem like no member of, what do the challenging matter to ME? Who keeps quiet, agrees with me!
Greet’s from Yours, faithfully,
J.A., to be continued.
I am supervising Galois Theory and found your site…hopefully my students don’t. Do you always answer questions or do you often give hints? I would prefer that if my students stumbled upon your site…they would not have found someone to do their work but rather another resource to help them learn the material! But in the end, it is your site and your call 🙂
Cheers,
–tc
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Could you help me with this very easy problem but with which I am having troube: prove that the single transcental extention of the field of the rational numbers is not normal? Q(t):Q is not normal?
Thank you for the help.
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