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If $p$ is a prime, prove that $x^{p-1}-x^{p-2}+x^{p-3}-cdots-x+1$ is irreducible over $Q$....

Question

If $p$ is a prime, prove that $x^{p-1}-x^{p-2}+x^{p-3}-cdots-x+1$ is irreducible over $Q$.

If $p$ is a prime, prove that $x^{p-1}-x^{p-2}+x^{p-3}-cdots-x+1$ is irreducible over $Q$.



Answers

Proof Prove that you can find a polynomial $p(x)=A x^{2}+B x+C$ that passes through any three points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),$ and $\left(x_{3}, y_{3}\right),$ where the $x_{i}^{\prime}$ s are distinct.

Okay, So this question we want to prove that if X squared is equivalent to one well, he This implies that X is equivalent to one or nice one. Well, so to do this, what will do it was subtract one to both sides. So we have X squared is minus one is equivalent to zero mud. Hey, so what this implies is that he is a divisor, or X squared, minus by one. Well, in fact, that extra admire spoke one p is divisor off extra mile one, and it's plus one now, since he is a crime, what? Since he has a point, we can use the fact that if p is divisor off A and B, this implies that he is a device off a or he is a device off. So we do the same result on he and we split up to develop division. So then we have p is a device off exit one by one, or he is a device off X plus one. No, Now you can sew in this question. We went from mod to devise er that we used a reverse equation to go back and we have extra mice where one is equivalent to zero month. He or X plus one is equivalent. She's there are model T one I won't make. Whenever we have exes equipment to one more. He oh X is equivalent to a modest one month pay and we have a result as you are.

So in this question, we want to show that if he is a crime, then such that piece on divisible by a then eight p months to is an inverse or a more. Hey, so despite definition off for my little theory, we have a two piece must buy. One is equivalent to one model p of assistance. Just spirits. That's all. L R T T O E. Yeah. Famous. It was here so we can decompose eight p months by one as being eight Pete miles bite you times by a and this is equivalent to one. Well, he and we already have a results because a times by eight he minds by choose gives you one month p so ends on therefore the inverse of a more he is this he wants to

Okay, so what this question? I want to show that X squared is equivalent to a more he that has no solution. Courts that to include your solutions. So he has no solutions. If a is not in the set off on critical issues Quadratic residues. Oh, he Now we then assume that a is Emiko Jared was juiced off peak. So then therefore there exists a solution equipment to a more he Now we also assume that why is a mother submission to this? So then because they're equal wasn't just said X squared is equipments to y squared Well, he mom so x squared. My wife sport is acquitted. That's 20 more he so therefore he's a divider off x squared minus y square. But we can factor exploit my wife's quit It's well hidden devices off X miles by Why x plus y So since peace device out these you have he is devising off x marks for why? Well, he is devising off X plus y therefore X minus y y month. Hey, is it to czar or he is equal base Plus for why mother he is your accuser. To combine these two results we have X is equivalent to Why? Well, he oh X is equivalent to negative one monkey. So hence we have two solutions. Hence we either have no solution. When a lot in the cortex residues Well, we have two solutions being positive. One negative ones.

Okay, so we want to show that if piers are crimes in, there are there are exactly t minus one divide about. So if he isn't a crime, then the greatest committed buys out with any interject smaller than peace. So that is the greatest common divisor off X. And he is eager to one for excellence in 212 boulevard to Pete months by one. Now, if G is a primitive route him achieve news. Okay, then we know that a is equal to G sweat. Jeet for Jason sixth, blah, blah blah. J T P must buy one on these. All have solutions to expert is equivalent to a more come in. So since this sequence, he contains P minus one, divided by two elements. So their API myself want Theo to numbers and he there are exactly k Mart's want to bottle up here.


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