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Find an irreducible polynomial f (z) Z,[z] degree Justify your answer.(b) Let root of the polynomial you found in the previous item in an extension of Z; What is th...

Question

Find an irreducible polynomial f (z) Z,[z] degree Justify your answer.(b) Let root of the polynomial you found in the previous item in an extension of Z; What is the cardinality of F = Zs(a)?With & as in the previous item find the minimal polynomial of 1/& and a2 +1 over Z;Determine the number of monic cubic irreducible polynomials over Z;

Find an irreducible polynomial f (z) Z,[z] degree Justify your answer. (b) Let root of the polynomial you found in the previous item in an extension of Z; What is the cardinality of F = Zs(a)? With & as in the previous item find the minimal polynomial of 1/& and a2 +1 over Z; Determine the number of monic cubic irreducible polynomials over Z;



Answers

Determine whether the given quadratic polynomial is irreducible. [Recall from the text that a quadratic polynomial $f(x)$ is irreducible if the equation $f(x)=0$ has no real roots] (a) $x^{2}+3 x-4$ (b) $x^{2}+3 x+4$

Hi everyone. So today we're going to be factoring a polynomial. So our polynomial today that we're going to factor in three steps. The first step is we're going to factor it, that we're going to find the factors that are irreducible over the rationals. So we're not going to have any like square roots or eyes in here for the first part. Then we're going to go a little further and we're going to find the factors that are irreducible over the reels. So we're not gonna have any eyes but we might have some square roots. And then finally for C, we're going to completely factor until you can't factor anymore. So to start off our equation that we're factoring today is f of X equals X to the fourth minus to execute minus three X squared plus 12, X minus 18. And if you're reading the book it gives us a little hint for the first part, it says that one of the factors is X squared minus 16. So we're just gonna start by pulling X squared or sorry, X squared minus six. We're going to pull X squared minus six out of there to see what we get. Mhm. So this hint was actually very useful. Once you pull X squared minus six out, you are left with X squared minus two X plus three. Which is irritating, irreducible over the rationals. So here's part A we found all the factors that are irreducible over rationals. Anything else would give us eyes or square roots. So that's as far as we can go. For part A now on to Part B. So in part B were allowed to have square roots because they're irrational but they're real. We're not allowed to have any eyes because those aren't real. So once you look at X squared minus six, you realize that you could do X plus route six X minus route six, you'll get the correct answer. However X squared minus two X plus three you can't do anything with because I would give you eyes. So this right here gives you part B f f x equals X plus route six times X minus route six times X squared plus minus two X plus three. Now on to Part C. And you have to remember that in part C. We are allowed to have anything that we want um square roots, eyes whatever. So part C is really about that second half of the equation. You can't do anything further with X plus Route six and X minus Route six. But you can break X squared minus two. X plus three. Down further to x minus one minus route to I And X -1 plus route to I. So part see your final answer is f of X equals X plus route six times X minus route six times x minus one minus root two's I, and at times x minus one plus route to I. And that is how you factor a polynomial in three different steps.

And the Lady Assembly or Mark you NFL fixes squared minus 16 because zero f discarded called 16 excessive Google plus or minus four. It has a really group. So it is traduced. Super okay, really is about. And to be part is a square plus 16 equals it'll next square It was minus 60 Exit Koldo Messer minus for I Best story a route, nor your work. And it is Iranian simple. It's ideal city.

Jim Book World Ex Krampus Jeanette's My This It's a longer this pragmatic one. This is irreducibly Super Mac. Really dream Check over a full face seepage Israel as you, the Utes. Most of the challenge here to do Is it violence doctrine? It was getting rid of loss. It's thing we can reduce If I was full sold this place. You notice this whole eggs? No, You six notes. Hey, thank you. Refugee objects and forced to a drum. Still big excuse for And I was form from last bedrooms. We could exist. Vince equals X zinc. Young column Momo's You see guys, they all sorts wells really groups we don't school. Therefore, clouds is not. Then when he was a good and then we make it Whose father they have a born woman extrapolate streets Smokeless sheet. It's yes, they've been Teoh so we can introducing the pipe activities The phobia. Use the district 12 So you just want Eugene Small. The formal mandates that his B it's That's a music video this DZ being so exposed You see him? It's the distance things here to you guys. It's full franks. Yes, you will be with chips to this case. Music will to be sly was nine minus old injury. See, it's a swell, Vincent stated. Well, nothing normal. 60. Just you guys them I say you won't guys. It still freaks as they move you towards you. Nice hole. If your faults use thinking it is.

You in function is X square plus three years minus four equals zero. So hey, it's foods would be later Exit girdle, one comma minus four. So these are every year roots created, so it is already radio simple again. So it is already assembled and be part of this. It's a square plus three years plus four equal to zero care. If we find the roads from self minus three bedroom, that sort of minus Group seven by a group. These are more Norio moves. These are imaginary, so including listen, but is idealism, but


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