Recent content by curiousmuch

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    Odd Degree Finite Field Extensions and Equality of Adjoined Elements

    Thanks for the help, but sorry we haven't learned about the "tower law." Can you help me with the part where you said that [F(a):F(a^2)]=2 if not 1.
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    Odd Degree Finite Field Extensions and Equality of Adjoined Elements

    Homework Statement Let F be a field, and suppose that alpha is algebraic over F. Prove that if [F(alpha):F] is odd, then F(a^2)=F(a). {For those unfamiliar with notation [] denotes degree of extension and F(alpha) means F adjoined with alpha.) The Attempt at a Solution Since [F(alpha):F]...
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    What are the ideals of Z mod 18Z?

    Homework Statement Find all ideals I of Z mod 18Z. Then find what (Z mod 18Z)/I is isomorphic to for every ideal I. The Attempt at a Solution We know that the whole ring and {0} are ideals. since Z/18Z is not a field there are more. So are Z/nZ where n is a divisor of 18, all of them?
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    Proving converse of fundamental theorem of cyclic groups

    Homework Statement If G is a finite abelian group that has one subgroup of order d for every divisor d of the order of G. Prove that G is cyclic. Homework Equations The Attempt at a Solution
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    Proving H is Cyclic: Finite Abelian Group

    Homework Statement Let H be a finite abelian group that has one subgroup of order d for every positive divisor d of the order of H. Prove that H is cyclicHomework Equations We want to show H={a^n|n is an integer}
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    Finite Commutative Ring: Proving Integral Domain w/ No Zero Divisors

    Homework Statement Show that a finite commutative ring with no zero divisors is an integral domain (i.e. contains a unity element) Homework Equations If a,b are elements in a ring R, then ab=0 if and only if either a and b are 0. The Attempt at a Solution I've been trying to use...