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Calculate each of the following.
Make sure that you have a field extension.
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Calculate each of the following.
Make sure that you have a field extension.
Calculate where
What is the lattice of subfields for
Let be a zero of over Construct a finite field of order 8. Show that splits in
There are eight elements in Exhibit two more zeros of other than in these eight elements.
Construct a finite field of order 27.
Find an irreducible polynomial in of degree 3 and show that has 27 elements.
Prove or disprove: is cyclic.
Factor each of the following polynomials in
(a) (c)
Prove or disprove:
True.
Determine the number of cyclic codes of length for 7, 8, 10.
Prove that the ideal in is the code in consisting of all words of even parity.
Construct all BCH codes of
length 7.
length 15.
(a) Use the fact that
Prove or disprove: There exists a finite field that is algebraically closed.
False.
Let be prime. Prove that the field of rational functions is an infinite field of characteristic
Let be an integral domain of characteristic Prove that for all
Show that every element in a finite field can be written as the sum of two squares.
Let and be subfields of a finite field If is isomorphic to show that
Let be fields. If is a separable extension of show that is also separable extension of
If then
Let be an extension of a finite field where has elements. Let be algebraic over of degree Prove that has elements.
Since is algebraic over of degree we can write any element uniquely as with There are possible -tuples
Show that every finite extension of a finite field is simple; that is, if is a finite extension of a finite field prove that there exists an such that
Show that for every there exists an irreducible polynomial of degree in
Prove that the given by is an automorphism of order
Show that every element in can be written in the form for some unique
Let and be subfields of If and what is the order of
Let be prime. Prove that
Factor over
If is the minimal generator polynomial for a cyclic code in prove that the constant term of is
Often it is conceivable that a burst of errors might occur during transmission, as in the case of a power surge. Such a momentary burst of interference might alter several consecutive bits in a codeword. Cyclic codes permit the detection of such error bursts. Let be an -cyclic code. Prove that any error burst up to digits can be detected.
Prove that the rings and are isomorphic as vector spaces.
Let be a code in that is generated by If is another code in show that if and only if divides in
Let be a cyclic code in and suppose that where and Define to be the matrix
and to be the matrix
Prove that is a generator matrix for
Prove that is a parity-check matrix for
Show that