# LARGE FUNDAMENTAL UNITS AND THE MONOMIAL NORM EQUATION

- Author:
- REITER, CLIFFORD ARNO
- Physical Description:
- 116 pages
- Additional Creators:
- Pennsylvania State University

- Summary:
- This thesis is motivated by Yamamoto's theorem that gives an asymptotic lower bound on the fundamental units of certain sequences of real quadratic fields. His theorem requires infinite sequences of real quadratic fields where primes p(,1),...,p(,n) are known to split into principal ideals in the ring of integers of the field. One concludes that log (epsilon)(,d) > c(log D(,d))('n+1) where (epsilon)(,d) is the fundamental unit, D(,d) is the discriminant and (SQRT.(d)) is in the sequence.

We generalize this in two directions. We say that a(,1),...,a(,n) (ELEM) are quadratically independent if.

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).

The first generalization replaces the primeness of p(,1),...,p(,n) by the quadratic independence of a(,1),...,a(,n) (ELEM) . The second generalization gives effective bounds while the complexity remains the same as in Yamamoto's result. This allows us to make specific statements about the size of (epsilon)(,d) in particular fields and allows us to avoid Yamamoto's hypothesis of having an infinite sequence of fields.

Yamamoto gives an example where n = 2. We notice that his sequence really uses the fact that D = (qx + p + 1)('2) - 4p = (qx + p - 1)('2) + 4qx with x = p('k). We pursue this idea with the goal of extending it to n (GREATERTHEQ) 3.

The monomial norm equations are A(,i)('2) - B(,i)('2)D = c(,i)x('e(,i)) where A(,i), B(,i), D (ELEM) {x}, e(,i) (ELEM) with 1 (LESSTHEQ) i (LESSTHEQ) n, and for c(,1),...,c(,n) (ELEM) quadratically independent. When these are over and we have one prime norm, p, then we can let x = p('k) and see that (epsilon)(,d) must be large, as in our effective theorem, for d = D(p('k)).

We develop the theory of these equations and get several strong necessary conditions. We also develop collections of APL programs that allow us not only to generate examples of these monomial equations, but also to check whether the example has other easy monomial norms.

This class greatly broadens the set of real quadratic fields which are known to have large fundamental units in this sense. - Dissertation Note:
- Ph.D. The Pennsylvania State University 1984.
- Note:
- Source: Dissertation Abstracts International, Volume: 45-06, Section: B, page: 1805.

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